Laser wakefield acceleration (LWFA)1 is characterized by a high acceleration gradient that is able to generate high-energy electrons within a short distance, promising the possibility of constructing compact particle accelerators. State-of-the-art laser wakefield accelerators are able to produce acceleration gradients of about 40 GeV/m, accelerating electrons to the order of 10 GeV within tens of centimeters using a 850 TW laser.2 Recent 10–100 PW laser systems3 are pushing LWFA forward toward the high-energy frontier. In general, higher laser power permits a longer accelerating distance and higher electron energy, i.e., ΔE_k ∼ P^1/3, where ΔE_k is the energy gain and P the laser power.4 These 10–100 PW laser facilities will raise the energy limit to the level of 100 GeV within a single stage, which can be boosted to the TeV level with the help of multistage LWFA acceleration.5 However, LWFA in the blow-out regime requires that the laser field strength be slightly above the relativistic threshold to maximize the acceleration energy.6,7 For a typical 1 PW laser pulse, the optimal spot size will be a few tens of wavelengths, resulting in more than 10 m of focal length. For LWFA systems, the focal length could be the longest part of the system and cannot be reduced with ordinary optical components. By considering the energy gain of an accelerator over the specific system size, the effective energy gain will then be 10 GeV in 10 m, canceling out the compactness of LWFA in plasma. For 10–100 PW laser pulses, the size of the focusing system could be several hundreds of meters and the acceleration per system length will be even lower. The extremely long focal length hinders the application of high-power lasers to LWFA and multistage LWFA toward 100 GeV, since the actual size of the whole system is beyond the capabilities of many facilities.

It is therefore crucial to shorten the optical length for LWFA if the system is to remain compact and capable of multistage acceleration.8,9 Owing to the damage threshold of the amplification media of the laser system, the diameter of the laser spot before the final off-axis parabola (OAP) is large for high-power lasers, limiting the possible reduction in focal length.

Here, we propose using a micro-plasma parabola (MPP) to transform a tightly focused pulse to a quasi-plane-wave beam of spot size suitable for LWFA. The size of the focusing system can be reduced to a few meters even for 100 PW lasers, which is one order of magnitude smaller than an ordinary focusing system. An MPP positioned behind the focus of an ordinary OAP of small f-number reflects laser pulses of small spot sizes to a quasi-plane-wave that is used as the driver pulse of the LWFA stage. To demonstrate the scheme, we carry out three-dimensional (3D) particle-in-cell (PIC) simulations with a 1 PW laser pulse. We find that the laser pulse reflected by an MPP produces an accelerating wakefield over the whole plasma length. The field structure and the electron energy gains are consistent with the case when the laser pulse is focused by an ordinary OAP. About 9 GeV electron acceleration is achieved within a total of 1 m optical length when the focal length of the OAP is less than 1 m and that of the MPP is about 675 μm, one order of magnitude lower than that of an ordinary focusing system. The scenario, when extended to 10–100 PW lasers, can reduce the optical length to a few meters, enabling compact LWFA systems toward 100 GeV.

In our scheme, the reflection of a tightly focused laser pulse by the MPP is simulated via a PIC method in 3D space, with the simulation being carried out using the EPOCH code.10 The LWFA stage is then simulated in quasi-cylindrical r–z coordinate via the FBPIC code,11 on the basis of the pulse reflected off the MPP, which significantly reduces the simulation time and makes it possible to simulate the 15 cm-long acceleration stage.

The 3D simulations are carried out in a 80 × 200 × 200 μm³ box with cell size of 0.05λ × 0.2λ × 0.2λ, where λ = 800 nm is the laser wavelength. Simulation of a 1 PW laser with large spot size is challenging. Numerical convergence of the chosen grid size is validated with a 100 PW laser in 3D and 1 PW in 2D (see the supplementary material). The MPP is placed at the left boundary of the simulation box with eight macro-electrons and one macro-proton (mobile) in each cell with a density 20n_c, where n_c is the critical plasma density. It is worth noting that the ion motion can significantly influence the electron radiation.12 For off-axis angle θ, the MPP is rotated by θ along the z axis, so that the reflected laser pulse propagates along the x axis, as shown in Fig. 1. The circularly polarized (CP) laser pulse is injected from the boundary of the simulation box with tilted angle θ. The 800 nm laser pulse is focused at 675 μm away from the MPP with a field strength a₀ ≈ 43 and a beam waist w₀ = 4 μm at e⁻¹ of the Gaussian profile, where a₀ = eE₀/mcω is the normalized field strength, with E₀ the peak electric field amplitude, m the electron mass, c the speed of light, and ω the angular frequency of the laser. The laser power peaks at 1 PW with a duration of about ten laser cycles, and the pulse energy is about 28.7 J measured in the simulation.

Considering the simulation efficiency, the electron density of the MPP is set to 20n_c ≈ 3.48 × 10²² cm⁻³ and its off-axis angle θ = 0.15 rad. The plasma density of 20n_c is sufficient to simulate the reflection of high-intensity lasers without requiring high spatial resolution. In fact, the plasma density depends on the ionization level of the target and the laser intensity. As the target becomes ionized above n_c, the laser is reflected, and a higher density will not fundamentally change the reflection. Larger off-axis angles do not change the interaction picture, but may need high resolution along the transverse directions.

The reflected pulse in the 3D simulation is then transformed into cylindrical coordinates via azimuthal Fourier decomposition.13 First, the electromagnetic field components F(x, y, z) in the Cartesian grid are converted to F(z, r, θ). The latter can be expressed by summation of the Fourier components: Fz,r,θ=∑mFmz,rexp−imθ, where Fm are the Fourier components and m is the mode number. The Fourier components are calculated via Fourier transformation Fmz,r=(1/2π)∫02πFz,r,θexpimθdθ. Such a conversion of fields is possible when the symmetry of the fields can be resolved by azimuthal Fourier expansion of e, which is accurate when the fields are highly symmetric along the θ axis. In our simulation, three modes are sufficient to model the reflected pulse.

The quasi-cylindrical simulation is carried out in a 4096 × 800 window with cell size of 0.05λ × 0.2λ in the z and r directions. Each cell is filled with 1 × 1 × 12 macroparticles in the z, r, and θ directions. The above-calculated Fourier components are loaded into the simulation window as the initial conditions.

We use a plasma channel14 to guide the laser pulse for long-distance acceleration.2,15 The density profile of the channel is expressed as ne(r)=n0+r2/(πrewm4), where n₀ is the central electron density, r_e ≈ 2.8 × 10⁻¹⁵ m the classical electron radius, and w_m the matched spot size.

For demonstration of the acceleration capability, we carry out LWFA simulations for both the reflected pulse and the pulse injected from simulation boundary, i.e., without reflection by the MPP. The simulations of the reflection by MPP are presented in Sec. III C. The laser pulses are injected into the plasma channel with central density n₀ ≈ 2.4 × 10¹⁷ cm⁻³ and a matched spot size w_m ≈ 43 μm. The injected pulse has field strength a₀ = 4, i.e., a₀ ≈ 2.83 in the y and z directions, is circularly polarized (CP), propagating along x, and has a peak power of 1 PW. We select 1 PW owing to computational constraints, since a higher laser power would increase the spot size, which is challenging to simulate in both 3D and quasi-cylindrical coordinates. The LWFA process is simulated via FBPIC11 in r–z coordinates with the reflected pulse as the initial condition. The simulation details are as described in Sec. II.

Figure 1 compares the proposed method with ordinary focusing systems. In the proposed setup, an incident 1 PW laser pulse is tightly focused by the OAP within 1 m, and this is followed by the MPP transforming the focused pulse into a plane-wave pulse that can be injected into the plasma channel within a few centimeters, depending on experimental conditions. This essentially focuses the 1 PW laser to a large spot in a short range, as opposed to traditional systems, which require more than 10 m to focus. The subfigures in Fig. 1 show the laser spots at the entrance and snapshots of the acceleration fields (E_z in red and blue) and electron densities (gray). The proposed method generates results consistent with those of the ordinary focusing system in terms of the wakefield structure.

For a concrete comparison, the long-term evolutions of the on-axis E_z and electron density are shown in Figs. 2(a) and 2(b) for the reflected and injected pulses, respectively, representing the evolution of the bubble structure and the electron beam. The yellow trajectories indicate the electrons with high density, i.e., the tail of the bubble and the injected electron bunch. It can be inferred from the electron bunch trajectory and the evolution of E_z that the electrons experience similar acceleration gradients. The corresponding electron energy evolution and final spectrum are shown in Figs. 2(c) and 2(d). It can be seen that electrons injected at similar positions are accelerated to similar energies despite the different bubble evolution. For example, electrons injected at ct ≈ 60 mm are accelerated to E_k ≈ 6 GeV and those injected at ct ≈ 40 mm are accelerated to E_k ≈ 8 GeV in both cases, which can be inferred from the gray lines in Figs. 2(c) and 2(d).

The reflection induces slight modification to the laser pulse, which results in small differences in bubble evolution after long-distance acceleration, as shown in Fig. 2. While the bubble structure oscillates at the beginning in both cases, the reflected pulse from the MPP seems to induce a longer bubble oscillation time. Self-injection becomes less continuous as compared with the injected pulse case. The discontinuous injection produces several energy spikes, as evidenced in Fig. 2(c). However, the energy range covering 2.5–9 GeV and the beam charge of the highest-energy part from both schemes are quite consistent. In addition, the angular divergence is not affected by the MPP reflection in our modeling. These subtle differences can be attributed to the nonlinearity of LWFA in the blowout regime.6,7,16 The features of the reflected laser pulse will be further analyzed below.

According to the scaling law of LWFA,4 the electron energy from the matched condition at different laser powers is shown in Fig. 3. When an ordinary OAP focusing geometry is used, the size of the beam incident onto the OAP scales as w₀ ∼ P^1/2 and the F-number as F_# ∼ P^1/2, resulting in a linear scaling of focal length f ∼ P. Considering typical energy density thresholds achievable on the mirror surface, generating a 100 GeV electron bunch requires hundreds of meters of optical length. By using the proposed MPP, the F-number of the OAP can be fixed to, for example, F_# ≈ 6 in all scenarios in our modeling, and the scaling law becomes f ∼ P^1/2, which reduces the focal length by two orders of magnitude when generating 100 GeV electrons, as shown by the red line in Fig. 3.

Taking into consideration the optical length, the effective energy gain per system size reaches 10 GeV/m, which is comparable to the acceleration gradient in the plasma. This drops to 0.1–1 GeV/m for ordinary focusing systems. It should be noted that the focal lengths of the MPP are insignificant compared with those of an OAP for 1–100 PW lasers. Similar effects can be achieved with multiple plasma mirror (PM) pairs inserted into the focusing optical path to reduce the optical footprint by an order, increasing the acceleration gradient, but at the cost of energy absorption by the PM pairs. Therefore, the introduction of the MPP preserves the acceleration gradient of LWFA without compromising reflection efficiency when the focal length of the focusing system is reduced.

The geometrical arrangement when an MPP is used to reduce focal length is shown in Fig. 4. To achieve short focal lengths, the F-number of the first focusing mirror used to focus the incident laser can be fixed to a small value, which will generate a tightly focused laser spot at the OAP focus. The MPP is positioned at a distance z₀ behind the focus, with a laser spot size w₀. The focal length of the MPP should be chosen to match the radius of curvature R of the wavefront at the MPP and transform the focused laser to a plane-wave laser, i.e., with f ≈ z₀, as a consequence of which the radius of curvature of the MPP is approximately twice that of the incident wavefront. The F-number of the OAP is flexible, as long as the combined OAP–MPP pair transforms the incident laser to the desired spot size and wavefront curvature. In the considered 1 PW situation, we choose F_# ≈ 6, which will generate a laser spot with a₀ ≈ 43 and w₀ = 4 μm for an 800 nm laser at the first focus. To get a reflected pulse with a₀ = 4 and w₀ ≈ 43 μm, the condition z₀ ≈ f ≈ 675 μm is chosen. The chosen off-axis angle is θ = 0.15 rad for computational efficiency, as discussed in Sec. II.

In realistic situations, the prepulse or pedestals of the laser pulse may produce preplasma on the front surface of the MPP, which will influence the energy absorption by the MPP from the laser pulse. The role of this preplasma and its modeling will be discussed later in this subsection.

The reflection and propagation of the pulse are shown in Figs. 5(a)–5(c). During the reflection, as shown in Fig. 5(a), the field strength is periodically modulated along the y direction by the interference between the reflected and incident pulse due to the extra optical path induced by the curved surface. On the other hand, at a field strength a₀ = 4, ion motion is insignificant and no visible distortion of the MPP is observed in Fig. 5(a). For heavier ions other than protons, their motion and the distortion of the MPP will remain negligible. After the pulse is reflected and the simulation window starts to move, slight modifications and noises induced by laser–plasma interaction can be observed behind the pulse in Fig. 5(b), which become absent after propagation through ct = 500 μm, as shown in Fig. 5(c). It should be noted that the density distribution and laser pulse are rotated by an off-axis angle θ = 0.15 rad, and so the reflected pulse propagates along the x direction.

During the laser–plasma interaction, high-order harmonics will be generated via the relativistic oscillating mirror mech-anism.18,19 Since high-order harmonic generation (HHG) is usually utilized in radiation sources,12,20–23 the generated harmonics could either influence the LWFA stage or directly lower the reflection efficiency of the base frequency. According to our modeling, as shown by the normalized frequency spectra in Fig. 5(d), the field strengths of the second- and third-order harmonics are less than 0.1 of the incident pulse, which is negligible in the context of LWFA and so the reflection efficiency of the base frequency remains high. This is because the high-order harmonic efficiency is suppressed for CP lasers.24 The overall reflectivity reaches 93.4% for a CP laser. However, for s/p-polarized lasers, the strength of the third-order harmonic can reach 0.2 of the base frequency, which could potentially affect LWFA stage and lower the reflection efficiency. The reflectivity of the LP laser drops to around 73% for both s- and p-polarizations in our modeling, owing to stronger high-frequency longitudinal oscillation of electrons in laser fields. The role of HHG in LWFA remains to be investigated for LP lasers. Therefore, focusing of LP PW-class lasers with the proposed MPP comes with an energy loss of about 30%, which requires a trade-off between long focal length and reduced laser energy when LP lasers are being reflected. Thus, in the scope of this study, only CP lasers are considered in the LWFA stage.

The role of preplasma is now investigated. In the case without preplasma, the reflected pulse and the transverse profile are shown in Figs. 6(a) and 6(b) and quantified in Fig. 6(c) in terms of the transverse profiles of the electric fields and the transverse phase relative to the pulse center. The relatively flat phase curve indicates that the wavefront is nearly planar. But the lowered field profile (solid red/blue) indicates that part of the pulse energy is lost after the reflection, which is 83.6% of the incident pulse, i.e., 24.0 J, owing to the absorption and heating of the electrons.

However, as already mentioned, in more realistic situations, the prepulse or pedestals of the laser pulse may produce preplasma on the front surface of the MPP, influencing its absorption of energy from the laser pulse. Therefore, an exponentially distributed preplasma of exp(−x′/l_pre) is added to the MPP surface, where x′ is the coordinate vertical to the rotated MPP and l_pre is the scale of the preplasma.17 In our modeling, we choose l_pre = 0.1 μm, which is a typical situation that can be realized by coating the mirror with CH material and by using a well-controlled prepulse. We note that the reflectivity can be boosted to 93.4% in the presence of preplasma with l_pre = 0.1 μm, but at the expense of a modification of the pulse profile, as shown in Figs. 6(d)–6(f), where the squeezed transverse profile and convex phase indicate that the laser is more focused than in the absence of a preplasma. This is because the presence of preplasma amplifies the denting of the MPP surface. The denting becomes more significant where the field strength is higher,25 which shortens the focal length. In our modeling, a larger l_pre will further amplify the denting effect and degrade reflectivity. In fact, the formation and evolution should be well controlled by adjusting the strength and delay of the prepulse whenever a PM is utilized.25

To mitigate the modification, we adjust the focal length of the MPP to f = 750 μm from the designed f = 675 μm. The transverse profile and the relative phase are recovered, as shown in Fig. 6(i), where the pulse profile is closer to the expected Gaussian profile (dashed gray line) and the phase near the center is as flat as in Fig. 6(c), indicating that the wavefront is almost planar and is suitable for the LWFA stage. This pulse is utilized as the driver pulse in the LWFA simulation described in Sec. III A. However, since the denting of the preplasma is intensity-dependent, the phase in the region r > 50 μm appears under-focused, which could modify the injection in the LWFA stage, as shown previously. As a result, by increasing the focal length in the presence of preplasma, the MPP is able to reflect the incident laser pulse with higher reflectivity without significant distortion of the wavefront.

It should noted that while flat PMs have been frequently employed to reflect ultra-intense laser pulses26,27 and improve beam contrast,26 PMs with curved surface have also been experimentally demonstrated to further enhance the focal intensity,17,28–32 which proves the feasibility of the proposed MPP. Similar to flat PMs, the MPP can be positioned much further from the laser focus with appropriate curvature matching that of the wavefront, reflecting the pulse at much larger spot size and lower field strengths. At lower intensities (<10¹⁶ W/cm²), PMs can be highly reflective even when the prepulse is not well controlled.25,33–35 However, such a setup is not computationally tractable owing to the large spot size. Therefore, for demonstration of the proposed MPP, the simulations are constrained to reflections at a spot size w₀ ≈ 43 μm and a field strength a₀ = 4.

With regard to manufacturing considerations, in the investigated geometry, the MPP structure can be manufactured via a 3D printing technique36 or from a rotating liquid that forms a parabolic surface (the normal direction is along that of gravity, which requires extra consideration in the experimental setup).37 On the other hand, the MPP can be replaced by an ellipsoidal PM (EPM).28,30 The foci of the ellipsoid form a focus-to-focus imaging system in which the spot size is magnified by β/α, where β and α are the distances from the foci to the reflection point of the EPM.38 The EPM can be manufactured on a macroscopic scale by tuning α, β, and the ellipticity of the EPM. Besides, the proposed method could benefit from the development of high-power EUV sources, which would enable the generation of liquid microdroplets with high stability39 in an approach that has been well tested in the chip fabrication industry. Such a droplet can be positioned before the OAP focus, with its curvature matching that of the wavefront of the focusing laser.

In terms of laser pointing stability, since the plasma channel can be situated much closer to the MPP, typically just a few centimeters away, positional jittering can be effectively managed. In contrast, conventional systems place the plasma channel tens to hundreds of meters away from the focusing mirror. For instance, a 1 μrad angular jittering of the first OAP mirror results in only about 10 μm of positional jittering when the plasma channel is 1 cm away from the MPP. However, in conventional systems, the same angular jittering would cause positional jittering of tens to hundreds of micrometers at the plasma channel, owing to the longer focal length, which extends up to hundreds of meters for higher-power lasers. In terms of longitudinal jittering of the focal point, if the focal point fluctuates within the Rayleigh range, the focal point of the reflected pulse also fluctuates around its Rayleigh range, according to Gaussian beam transformation. In other words, the insertion of an MPP does not require higher focusing stability of the focusing system.

For single-stage acceleration beyond 100 GeV, a long pulse is required, owing to the etching effect in LWFA.4 Ion motion can be significant with long pulses and a field strength of a₀ = 4. In our evaluation (see the supplementary material), the tail of the long pulse is significantly modified for fully ionized proton targets. This effect can be reduced by using partially ionized high-Z targets.

The MPP proposed here effectively transforms a 1 PW laser pulse focused by a short-focal-length OAP to a pulse with long Rayleigh length, useable for LWFA. The pulse reflected by the MPP successfully generates a 9 GeV electron bunch in the subsequent LWFA stage. Although slight modifications are introduced into the reflected pulse, the acceleration gradient and bunch emittance are similar to those in an ordinary focusing system. The proposed method essentially provides a new option to reduce the focal lengths of 1–100 PW laser systems when a large spot size is required as in LWFA. The compactness of an LWFA based on PW-class lasers is significantly improved, paving way for multistage acceleration toward 100 GeV and even TeV electrons.

The supplementary material encompasses the numerical convergence of the simulation under higher grid resolutions and the long-pulse effect.

Acknowledgment. This work is supported by the National Key R&D Program of China (Grant No. 2022YFE0204800), the National Natural Science Foundation of China (Grant Nos. 12388102 and 11935008), the CAS Project for Young Scientists in Basic Research (Grant No. YSBR060), the China Postdoctoral Science Foundation (Grant No. 2022M713258), the Shanghai Science and Technology Development Foundation (Grant No. 22YF1455100), and the International Partnership Program of the Chinese Academy of Sciences (Grant No. 181231KYSB20200040).