Dynamic shock compression is a pivotal method for the generation of warm and hot dense matter akin to the extreme conditions observed in planetary interiors, supernovae, and astrophysical jets. In the laboratory, a variety of large-scale facilities are used to drive dynamic shock compression, including gas guns,1–3 pulsed power systems,4–9 and high-energy lasers.10–15 Gas guns apply high-velocity gas to drive a projectile up to ∼16 km/s, thereby impacting the target to generate high pressures (∼TPa).2,3 Pulsed power systems4–6 are another way to generate high pressures. They can produce intense short-duration currents, ∼100 ns in duration, with peak values reaching into tens of mega-amperes.7–9 When these currents flow through a target, a resultant J × B force can be generated. Pulsed power systems have a wide range of applications, including the study of equations of state (EOS) by accelerating flyer plates to speeds of up to 45 km/s, which can generate pressures up to ∼3 to 4 TPa,16–20 quasi-isentropic compression,21,22 and generation of x-rays and investigations of fusion by the Z-pinch method.23–26 Nanosecond kilojoule-level high-energy lasers can also generate TPa pressures on targets by ablative-driven shocks or ramped compression.12–15 Recent advances in femtosecond high-intensity lasers offer an alternative way to generate short-pulse currents with duration potentially much shorter than the usual unstable magnetohydrodynamic (MHD) modes.27 When a femtosecond relativistic laser beam is incident on a solid target, energetic hot electrons with kinetic energy exceeding MeV are generated through the Lorentz force.28–30 The subsequent escape of hot electrons from the target leads to the formation of a surface return current31–35 and a corresponding tens of kilotesla magnetic field.36 Previous research on the return current has largely concentrated on the conventional Z-pinch effect, which employs the substantial J × B force to continuously drive the target to high densities and temperatures with significant instabilities.31,34,37,38

By contrast, our focus here will be on utilizing the transient nature of the return current on low-Z hydrogen jet to high-Z metallic wires to initiate converging shocks. We find that the return current lasts within a 100 fs timescale and is constrained within the target’s skin depth and hence allows the bulk of the target to maintain a relatively cold state. The return current will generate a transient J × B force [see the purple cylinder in Fig. 1(a)], as well as rapidly increasing the target surface temperature via Joule heating [see the red cylinder in Fig. 1(a)], and the resultant transient pressure gradient will generate strong shocks [see Fig. 1(b)]. Plasma ablation will further compress the target. Consequently, the converging shock benefits from the transient current duration, reducing the need for complex designs to suppress the MHD instability39–44 in the traditional Z-pinch and to achieve symmetrically converging shocks. To study this, a comprehensive approach that combines particle-in-cell (PIC) simulations and hydrodynamic simulations is applied. The PIC method allows us to understand the mechanism of compression initiation, while the hydrodynamic simulations offer a robust approach to model the effect over longer times. We demonstrate that the dynamic shock compression initiated by the return current is a consequence of the interplay between magnetic compression and surface ablation. In radii of few micrometers in low-Z materials, the shock compression is dominated by the magnetic compression. A predicted rebounded shock wave in a 2.5 µm solid hydrogen jet is indirectly indicated experimentally via shadowgraph images.45 We scale the return current dynamic compression to materials with high Z and larger radii. It is found that surface ablation is the main mechanism of initiation of a convergent shock under such conditions. In a 10 µm copper wire, the initial thermal pressure is 38 times the magnetic pressure. The pressure at the target center when it is imploded reaches up to 125 TPa. A recent experiment at EuXFEL shows clear convergent shocks on the copper wires.46 These findings establish return current compression as a promising novel platform for future high-pressure physics research.

The structure of the remainder of this paper is as follows. In Sec. II, we present hydrodynamic time-scale images of short-pulse laser–wire target interactions. In Sec. III, we discuss the generation of the return current and its subsequent effect on the target, together with a multi-timescale analysis to examine the mechanism underlying the dynamic compression initiated by the return current. In Sec. IV, we present experimental data on the evolution of the plasma shadow diameter, as captured by shadowgraphic imaging techniques. The results might hint at a combination of surface ablation and the J × B force originating from a laser-initiated return current. In Sec. V, we summarize the conclusions of this study.

A 3 J, 30 fs short pulse with an FWHM width of 3 µm is used in this study. As shown in Fig. 1(a), the laser-accelerated hot electrons (blue arrows) penetrating into the target cause the bulk return electron flow (dashed cyan arrows near the hot electrons). A portion of the hot electrons escaping the target cause the surface return electron flow (dashed cyan arrows on the target surface) and surface return current (purple arrows) in the skin depth. We divide the entire wire target into two regions. Region I is the laser spot region. Here, the incident laser (black arrow) directly irradiates the wire target. The laser energy is deposited into the target bulk directly or indirectly through return current Joule heating, inverse Bremsstrahlung absorption, and direct collisional heating by laser-accelerated electrons. In region II far away from the laser spot, the surface return current dynamics provides the main mechanism driving the plasma dynamics.

In the laser incident region I, as shown in Fig. 1(c), the dynamics are more complicated compared with those in region II. Here, the rapid laser energy deposition leads to a high-temperature hotspot, as illustrated by the red semicircle in Fig. 1(a). This hotspot generates a spherical shock wave propagating into the wire. Additionally, surface ablation can also generate a convergent shock. The density evolution in region I will be the result of these shock wave interactions. On the basis of the differences in laser contrast, the laser spot region can be heated to become transparent before the main pulse, depending on the target density and thickness.47–49 In those cases, the shock wave structure will not be as significant as in Fig. 1(c).

In region II, the surface return current can introduce a strong ablative force [red cylinder in Fig. 1(a)] by Joule heating and J × B force [purple cylinder in Fig. 1(a)] to compress the target. The hydrodynamic description of the compression process can be seen from the target density projection depicted in Fig. 1(b). Here, a 10 µm diameter solid-density copper wire is irradiated by the short-pulse laser. Hot corona plasma expansion (white arrows) induces a converging shock wave that propagates inward (red arrows) into the uncompressed wire core. The orange dashed line at t = 3 indicates the implosion region of convergent shocks. Above the dashed line, the converged shock is about to implode, revealing a very high-density shock front. Below the orange dashed line, the rebound shock wave and a relatively uniform high-density core are created after the convergent shock implosion, with the rebounded shock wave moving outward. Surrounding the shock wave, an expansion corona plasma with lower density is observed.

Owing to the high density of the convergent shocks and the low density of the corona plasma, direct probing of the shock waves at optical wavelengths will be difficult. The corona plasma will prevent probing rays from going through the target. A promising alternative is to use the X-ray phase contrast image (PCI) method, as shown in recent work on the European XFEL.46,50 In this paper, we focus on region II. On the basis of analytical simulations in Sec. III and experimental data in Sec. IV, we will demonstrate the existence of a converged shock induced by the surface return current.

The generation of the return current is discussed in Sec. III A. The surface return current has two primary effects on a wire target: J × B force and Joule heating. These two effects are discussed in Secs. III B and III C, respectively. The converging shock propagation on a 100 ps time scale is presented in Sec. III D. A return current scaling law is introduced in Sec. III E to evaluate the convergent shock compression mechanisms under conditions where the laser beam is incident on high-Z and large-radius targets.

PIC simulations are a superior tool for investigating short-pulse laser–solid interactions. In this study, we apply PIC simulations to explore the surface return current dynamics in wire targets. Here, to investigate the surface return current generation process, we employ a PIC method using the 2D3V code PICLS51 and the 3D3V code PIConGPU.52 A relativistic binary collision operator51,53 is included to calculate electron and ion temperatures. The 2D and 3D PIC simulations are used in Sec. III A to demonstrate the existence of the return current. The 3D PIC simulations are used in Sec. III A to quantitatively calculate the return current intensity and verify Ampère’s law in a cylindrical coordinate system. The 2D PIC simulations with longer simulation time and larger simulation box are used to show the dynamic process of surface return current compression in Sec. III B.

For the hydrogen target, in the 2D PICLS simulations, the target is presumed to be fully ionized solid hydrogen with a density of 30nc800nm, where nc800nm=1.74 × 10²¹ cm⁻³ is the plasma critical density corresponding to an 800 nm wavelength laser. The input laser is a Gaussian wave in time and space with a peak intensity of 5 × 10²⁰ W/cm² and a wavelength of 800 nm. The FWHM duration of the laser is 30 fs and the FWHM spot size is 3 µm. The laser propagates along the x direction and is polarized in the y direction. The simulation time t = 0 corresponds to the laser field on the target surface with peak intensity. Field and particle-absorbing boundary conditions are applied to simulate an infinite-length hydrogen jet. The simulation results are shown in Fig. 2. The comprehensive setup details of the 2D and 3D PIC simulations are given in Appendix A.

In this subsection, we analyze the initialization of the surface return current. We adopt the concept of the plasma β, which serves as a criterion to determine whether the plasma compression is dominated by a magnetic field or thermal pressure. The plasma β is defined as β = P_T/P_B, where P_T = n_ek_BT_e is the surface ablation plasma pressure and PB=Bθ2/2μ0 is the surface magnetic pressure. Here, n_e is the electron number density, k_B is the Boltzmann constant, T_e is the maximum electron temperature on the target surface, B_θ is the maximum toroidal magnetic field generated by the surface return current [see Fig. 1(a)], and μ₀ is the vacuum permeability. The initial step involves the derivation of the surface return current from the PIC simulations in order to compute the plasma β.

Figures 2(a) and 2(b) show the distributions of current density and magnetic field, respectively, in the PICLS simulations at 160 fs, illustrating the propagation of a current pulse on the surface of the plasma column. A zoom-in of the current density distribution around y = 40 µm is shown in Fig. 2(c) to address the distribution of the hot-electron current. The spatial extent of this current pulse is ∼30 µm. As the current pulse propagates approximately with the speed of light,54 the temporal duration of the return current pulse is estimated to be around 100 fs. Figures 2(d) and 2(e) show lineouts of the current density and magnetic field at y = 40 µm (40 µm from the laser spot center), where the current density and magnetic field on the plasma reach their peak intensity. The average peak current density is 0.28 MA/μm², and the average peak magnetic field is 19 kT. Similarly, the results of the 3D PIC simulations are presented in Fig. 10 in Appendix A. In this scenario, the peak current density is considered to be at a distance of 5 µm from the laser spot center. The corresponding current density and magnetic field are 0.9 MA/μm² and 46 kT, as shown in Figs. 10(e) and 10(f).

The surface current has several components: the escaping hot-electron current I_es, which initiates the generation of surface return current in region I and is negligible when surface return current propagates along the wire surface in region II, the forward-propagating laser-accelerated electron current I_h outside the plasma column, and the backward surface return current I_re in the skin depth of the target, which neutralizes the forward laser-accelerated electron current and escaping electron current. Thus, we have the equation I_re ≃ I_h. Here, the forward current is the current moving along the wire surface away from the interaction area, and the return current is moving toward the interaction area. The forward electron current and backward surface return current are in different layers along the radial direction. The interface of these two layers is around the plasma surface (r ∼ 2.5 µm in the 2D PIC simulations). The current is calculated by integrating the current density over the radius radially, as depicted in Eq. (2) below.

Figure 3(a) shows the current at an offset of 5 µm to the laser spot from the 3D PIC simulations (see Appendix A). The backward surface return current I_re, which is distributed in the 0.1 µm skin depth, increases dramatically to 210 kA at the plasma surface. A decrease in the integrated current density is observed when the radius exceeds 1 µm, indicating the existence of the forward laser-accelerated electron current I_h. The current drops to zero when the radius is 4 µm, showing that I_h and I_re are approximately equal to each other. The time variation of the simulated backward surface return current I_re at y = 5 µm as shown in Fig. 3(b). A backward surface return current with a duration of 100 fs is observed.

The magnetic field generated by the current can be deduced using the Biot–Savart law:Bθ(r)=μ0I(r)/(2πr),(1)withI(r)=∫0r2πrjydr,(2)where r is the distance to the target center, I(r) is the radial current distribution, and j_y is the current density along the y direction. B_θ(r) is compared with the magnetic distribution obtained from the 3D PIC simulation. B_θ(r) and the magnetic field from the 3D PIC simulation are shown by the solid and dashed curves, respectively, in Fig. 3(a). The maximum difference between the two magnetic fields is within 1%, demonstrating that the surface magnetic field (r ∼ 1 µm) is generated by the surface return current I_re and drives the Z-pinch effect on the hydrogen plasma. Consequently, the surface return current can be utilized to analyze plasma surface heating and compression.

In this subsection, we analyze the magnetic compression initiated by the surface return current. In the 2D PIC simulations, the region offset by 60 µm from the laser spot center is specifically selected to investigate the pure surface return current-driven magnetic compression of the target. Figures 4(a) and 4(b) present lineouts of the electron density and electron temperature evolution in this area. Here, the bulk and surface return currents both contribute to the electron temperature.55,56 The surface return current heats the surface electron to 1.0 keV within the skin depth at 240 fs, after which the rate of increase in surface electron temperature rises slowly owing to the rapid decay of the surface return current after 240 fs. The bulk return current heats the bulk electron temperature from 350 eV at 240 fs to 1.1 keV at 480 fs. The magnetic field pressure and thermal pressure are shown in Fig. 4(c). Before 300 fs, the plasma β = P_T/P_B ≤ 1, and hence the transient magnetic pressure drives the density compression, up to a compression factor of 3. This mechanism is consistent with the traditional Z-pinch effect. The J × B force driver only lasts for 100 fs, and the following shock wave velocity can be assumed to be constant in a short time. Based on the compression wavefront at respective times of 240, 320, 400, and 480 fs, we calculate the inward velocity of the compression wave to be ∼800 km/s. The sound speed of the hydrogen iscs=γp/ρ=γZkBTe/mi≈100km/s,(3)where γ = 1.5 is the adiabatic index, p and ρ are the uncompressed hydrogen pressure and density, Z is the atomic number of hydrogen, T_e is the electron temperature of uncompressed hydrogen, and m_i is the ion mass of hydrogen. The magnetic field induced by the surface return current generates an initial inward shock wave with a Mach number of 8. Magnetic compression ceases beyond 300 fs, as indicated by a plasma β ≥ 1 in Fig. 4(c). This demonstrates that the Z-pinch effect only dominates for the duration of the current pulse.

In the specific region considered here, bulk return current heating reduces the pressure gradient between the plasma surface and the bulk, and so the generation of the shock due to surface return current heating is not visible. We aim to isolate a region where the plasma heating is solely attributable to the surface return current. However, this requires simulation of a target with a transverse dimension extending to several 100 µm, which is extremely challenging for the PIC method, owing to its great demands with regard to computational time. Consequently, to accurately estimate the compressive effects of the surface return current on the target, a method beyond the scope of PIC simulations is necessary.

In this subsection, we explore the heating effects of a pure surface return current on a 2.5 µm radius solid hydrogen target over a 100 fs timescale. The corresponding surface electron temperature influenced by this current can be deduced using the particle energy equation56,5732ne∂Te∂t=∂∂rKTe∂Te∂r+jy(r)2σTe,(4)where KTe is the thermal conductivity of cold electrons and σTe is the electrical conductivity. The right-hand side of this equation accounts for heat diffusion in the radial direction and the Joule heating due to the surface return current. The drag heating term in Eq. (20) of Ref. 56 is omitted because of the negligible hot electron current within the bulk of region II in comparison with the surface return current. The electron resistivity model and the heat diffusion coefficient are functions of temperature and density and are extracted from the SESAME equation of state.58

The distribution of current density in the radial direction is included in Eq. (4) to study surface return current Joule heating in the plasma skin depth. Here, the skin effect of the surface return current has to be considered. The skin depth at electron temperatures under 100 eV can be calculated as ϵ=2/σTeωμ0≈0.1µm, where ω = 2π/τ is the frequency of the surface return current, with τ being the duration of the current pulse, and σTe is the electrical conductivity of the hydrogen.59 Equation (4) is solved in a 1D cylindrical geometry with an initial electron temperature of 1 eV, and the corresponding distributions of electron temperature at 20, 50, and 100 fs are depicted in Fig. 5. As the return current propagates through the target, Joule heating of the surface plasma is observed. The peak temperature reaches 500 eV at 50 fs and then decreases to 300 eV at 100 fs owing to heat diffusion within the 0.1 µm skin depth layer. Heat diffusion beyond the skin depth layer in the radial direction appears negligible in Fig. 5. As a consequence, bulk electrons remain relatively cool at 100 fs. This differs from the PIC simulation in Fig. 4(b), where bulk heating of the electrons is observed. The plasma with high surface temperature is expected to generate several tens of TPa pressure, which can also contribute to the formation of converging shocks. This will be discussed in Sec. III D.

In this subsection, we combine the effects of Z-pinch and surface heating to explore convergent shock compression over 100 ps time scales. However, the magnetic pressure is still observed after 480 fs, as shown in Fig. 4(c), revealing the existence of a magnetic field. To estimate the influence of the remaining magnetic field on the convergent shock evolution on ps time scales, we utilize the magnetic Reynolds number, which gives an indication of the magnetic diffusion effect and magnetic freezing effect. The magnetic Reynolds number, which provides an estimate of the effects of magnetic advection relative to those of magnetic diffusion, is given by R_m = UL/η ≈ 9.2, where U ≈ 300 km/s is the characteristic velocity, L = 5 µm is the characteristic length, and η=(μ0σTe)−1 is the magnetic diffusivity. A magnetic Reynolds number R_m > 1 indicates that the magnetic field will be frozen in the ablation plasma and will have no influence on the shock wave evolution in the bulk plasma. Therefore, the magnetic diffusion effect is excluded from this study. Consequently, a hydrodynamic method can be applied.

We aim to investigate pure surface return current compression without bulk return current heating of the hydrogen. We assume that the bulk plasma infinitely far away from the laser spot is cold, with an initial temperature of 1 eV, and that there is no decay for the surface return current density when the surface return current pulse propagates along the wire. The Z-pinch and surface heating effects provide the initial kinetic energy and thermal energy of the hydrogen plasma, respectively. Therefore, the electron temperature in Fig. 5 and the Lorentz force are the initial conditions of the hydrodynamic simulations. The radiation cooling of hydrogen at a 300 eV scale is 0.14 eV per electron and is excluded from our considerations.56 We solve 1D cylindrical, one-fluid, two-species (ion and electron), and two-temperature hydrodynamic equations using the FLASH code.60,61 The hydrogen SESAME equation of state58 is adopted.

The results of the simulations are shown in Fig. 6(a). It is evident that converging shock compression has three distinct phases. The initial phase features expansion of the hot ablation plasma into the surrounding space, along with the formation of a converging shock wave. This shock wave moves inward at an average velocity of 700 km/s during the initial 2 ps. The density compression factor in this period ranges from 4 to 5. As the shock wave moves inward, more mass is accelerated by the shock wave. A decrease in the ablation plasma’s temperature to 50 eV due to the expansion around 5 ps is also observed. Both factors contribute to the reduction in the shock velocity to roughly 400 km/s. The subsequent phase witnesses the converging shocks reaching the target’s center and collapsing around 8.5 ps. The density compression factor can reach a factor of 29, and the peak pressure can reach 100 TPa with 220 eV temperatures in a plasma core, as shown in Figs. 6(b)–6(d). The final phase is characterized by the converging shock waves rebounding after reaching maximum compression, and subsequently propagating outward. At around 40 ps, the reflected shock waves reach the interface of the plasma expansion, enhancing the expansion speed from 100 to ∼250 km/s. The corresponding plasma temperature is decreased to a few electron volts.

The simulation results show that the shock wave is initiated by the magnetic field compression and ablation plasma heated by the Joule heating associated with the return current. In the case considered here, magnetic field compression (plasma β < 1) is the dominant mechanism to initiate the shock wave. Owing to the short duration time of the return current, the energy can only be deposited in the skin depth of the target, resulting in a clean and symmetry-converged shock compression to the target to extremely high density and pressure. In Sec. III E, we extrapolate our theory to targets with higher atomic numbers Z and larger radii to explore the broader applications of dynamic compression steered by the return current methodology.

In this subsection, we extend the surface return current density to arbitrary atomic number Z and target radius to investigate the initiation mechanisms of convergent shocks via surface return current. The 2D and 3D PIC simulations show that the surface return current is quite similar for different target radii. Hence, we propose that the surface return current is governed mainly by the incident laser, with the return current duration and the corresponding skin depth remaining constant for fixed laser parameters. On the basis of Eq. (2), for different radius targets, the scaling of the surface return current density can be derived fromj1j2≈r2r1.(5)The corresponding magnetic field on the target surface can be derived fromB1B2≈r2r1.(6)From the 2D and 3D simulation results using wire radii of 2.5 and 1 µm respectively, it can be seen that the ratio of the maximum magnetic field on the plasma surface is B_2max/B_1max = 18 kT/46 kT ∼ r₁/r₂ = 0.4, which shows that the 2D and 3D PIC simulations are consistent with each other. A scaling law can be derived for various materials with different radii based on Eqs. (4)–(6). The heat transfer term in Eq. (4) is neglected, and the surface return current in a 2.5 µm radius hydrogen jet is taken as the input. The transient peak electron temperature on the plasma surface with a 30 fs, 5 × 10²⁰ W/cm² laser is given approximately byTe=2.2530nc800nmni0.42.5μmr0.8[keV],(7)where n_i is the number density of the target and r is its radius. The peak magnetic pressure, which varies with the target radius, is given byPB=101.62.5μmr2[TPa].(8)In the dynamics of a surface return current interacting with a wire target, the peak values of P_B and P_T are not reached simultaneously. Nevertheless, given the transient character of the surface return current, the plasma β remains a reliable metric to discern which mechanism is predominant in initiating a convergent shock within the wire target:β=PTPB=ZnikBTePB=Zni0.641.62.5μmr−1.2,(9)where n_i is the target atom density, with the unit of critical density corresponding to a laser wavelength of 800 nm, and Z is the target atomic number. It can be observed that the plasma β is proportional to the atomic number times the target atom density to the power of 0.6 times the target radius to the power of 1.2. The plasma β is plotted as a function of Z and target radius in Fig. 7(b). The two dashed contours correspond to β = 1 and β = 10.

Here, we use copper wire targets with radii of 10, 20, and 30 µm as representative examples. The corresponding plasma β values of 38, 87, and 142 indicate that thermal pressure, induced by the surface return current Joule heating, dominates the surface plasma pressure and initiates a convergent shock. Using the surface return current scaling law, we can estimate the peak surface return current densities as 90, 45, and 30 kA/μm² respectively. From Eq. (4), we can calculate the corresponding peak surface temperatures to be 300, 100, and 50 eV. Using the hydro simulation code FLASH with the copper SESAME equation of state,58 the peak pressures at the target center compressed by the surface return current are predicted to be 125, 5.3, and 1 TPa respectively. Figure 8(a) shows a plot of the copper density as a function of time and radius for a 10 µm radius copper wire. Convergent shock compression is observed. The convergent shock is imploded at about 550 ps. The peak pressure is 125 TPa, as shown in Fig. 8(b), where dashed line at ρ/ρ₀ = 4 divides the parameter space of into two regions. To the left is the principal Hugoniot region. Here the Hugoniot curve of copper is consistent with the experimental observations. For instance, a pressure of ∼2 TPa is achieved with a shock wave velocity of ∼22 km/s.62,63 On the right of the parameter space, the state is far from the principal Hugoniot curve. The compression ratio jumps from 4 to 11 owing to the reflection of the converging shock wave from the wire axis, which significantly reduces the volume of the target, leading to a much higher pressure being attained. This clearly shows the advantages of short-pulse laser-generated convergent shock compression. A similar pressure in copper can also be achieved in experiments with strong shock waves.64–66 A detailed theoretical study of the reflection of a converging cylindrical shock wave from the cylinder axis can be found elsewhere.67–69 We should note that in this calculation, we have ignored the radiation loss from the hot copper. On the basis of a previous study,56 the radiation loss is estimated to be less than 15% of the initial temperature. This can lead to a peak pressure decrease of around 10%. The radiation properties of hot dense copper can be found in Ref. 70. Figures 8(c) and 8(d) show the electron energy density and the surface return current density in a 10 µm width copper slab irradiated by a a₀ = 15, 30 fs, and 3 µm laser (the details of the setup of the PIC simulation can be found in Appendix A). The surface return current heating occurs in about 0.1 µm skin depth. Compression of the ion density is not observed during the first 100 fs after the peak laser pulse. The simulation results show that the thermal pressure is about ten times the magnetic pressure at 107 fs.

With the ideal gas equation of state, the transient shock velocity can be derived from the Rankine–Hugoniot relations:Us=U1+cs1+γ+12γp2p1−1,(10)where c_s is the upstream sound speed, with the electron temperature set to 1 eV (cold plasma), p₂ = P_B + P_T is the downstream pressure, γ = 1.5 is the constant ratio of specific heats, and U₁ is the upstream velocity, which is set to zero owing to a highly transient return current. The obtained plot of the shock velocity as a function of target radius and atomic number Z is presented in Fig. 7(a).

Our findings reveal that in low-Z and small-radius targets, the magnetic pressure stands out as the principal compression mechanism. Yet, the J × B force induces Magnetohydrodynamic Rayleigh–Taylor (MRT) instabilities on the plasma surface.37 Such instabilities disrupt the symmetry inherent to a convergent shock, thereby diminishing the attainable pressure—particularly in low-Z and small-radius configurations.71,72 Utilizing the established framework, we deduce the MRT instability growth perturbation amplitude ratio asη/r=η0rexp∫0tgkdt∝1rexp(r−3ρ0−1.5),(11)where η₀, g, and k are the amplitude, acceleration, and wavenumber, respectively, of the initial instabilities. On the basis of the above analysis, we propose the use of high-Z and larger-radius targets. In such configurations, the ablation plasma significantly leads to the initiation of clean dynamic shock compression, while, simultaneously, the occurrence of instabilities is reduced. The transient pressure can produce a shock with velocities ranging from 100 to 1000 km/s, substantially larger than the sound speed, for an atomic number Z up to 50 and a target radius up to 20 µm. This demonstrates that surface return current shock compression is a promising technique for generating transient high pressure on picosecond time scales.

In conclusion, we have initially employed a hydrogen jet because of its simple atomic structure and low density to investigate the initiation of a surface return current and a convergent shock wave through PIC simulations. Through these simulations and theoretical analyses, we have explored the interplay between the Lorentz force and Joule heating in initiating a convergent shock wave. The resulting return current and Lorentz force serve as initial conditions for hydrodynamic simulations aimed at studying the convergent shock wave on longer time scales. We have examined the dynamics of the surface return current for metallic wires with large radius and high atomic number using scaling laws. The results demonstrate the presence of a convergent shock wave for target radii up to 20 µm. To validate the theory, we now analyze hydrogen jet shadowgraph data to reveal the existence of a convergent shock wave.

As discussed in Secs. II and III, the converging shock compression initiated by the surface return current is characterized by three significant effects, namely, those due to a hot surface plasma, a cold bulk plasma, and a reflected shock wave. Each of these plays an essential role in understanding and analyzing the phenomena associated with convergent shock compression. These effects are reflected in the plasma density distribution, which can be probed by optical laser shadowgraphy.45,56,73,74 Experiments conducted at the HZDR 150 TW Draco laser facility and reported in Ref. 45 provide valuable insight into the occurrence of converging shock compression in hydrogen jets. In these experiments, a solid hydrogen jet with a radius of 2.5 µm was irradiated by a 5 × 10²⁰ W/cm² intensity, 30 fs short-pulse laser. Figure 9(a) and Fig. 12 (see Appendix C) present the shadow radius of the solid hydrogen taken from Fig. 5.3 (row 3) in Ref. 45 at various time delays relative to the pump laser. The differently colored curves represent the shadow radius at different distances from the laser spot. Here, we take the offset of 120 µm as a reference for the simulations. The bulk temperature at 120 µm is very similar to the initial condition of the simulation in Sec. III C (for details, see Appendix B). Ray tracing simulations connects the computational predictions with the experimental data,56,73 indicating the role of the surface return current in initiating converging shock compression on hydrogen jets.

As shown in region 1 in Fig. 9(a) and 9(b), a rapid shadow radius expansion rate within the initial 20 ps indicates the presence of a hot corona plasma. Assuming a constant critical density for a 515 nm laser probe, the hot ablation plasma is expected to expand at an ion sound speed of v=γkBTe/mi∝Te0.5. The expansion velocity can be estimated from the initial slope of the shadow radius. We use a shadow radius 120 µm offset to the laser spot, where the bulk heating is small. The slope can be calculated as 360 km/s. With a calculated temperature of 300 eV (see Fig. 4), the corresponding expansion velocity is 207 km/s, which is roughly consistent with the experimentally observed slope.

Transparency of the plasma is not observed in the shadowgraph image. This is distinct from the isochoric heating process.56 In the ray tracing simulations, we can identify the apparent difference between these two conditions. Figure 9(c) displays the shadow radius of adiabatically expanded plasmas with temperatures of 100, 200, and 300 eV. The expansion shadow radius for isochoric heating with electron temperatures 200 and 300 eV agree well with the experimental data in the first 20 ps. However, both shadow radius became transparent at 80–100 ps, indicating that the bulk plasma temperature is overestimated under the 200–300 eV isochoric heating assumption. When we decrease the initial temperature to 100 eV, transparency of the plasma is not observed, but the shadow radius in the first 20 ps are less than the experimental values, showing that the 100 eV temperature is underestimated. This comparison suggests that the bulk plasma temperature is significantly lower than that of the corona plasma. This is consistent with the surface return current heating mechanism described in Secs. II and III.

As shown in Fig. 9(a) and 9(b), the shadow radius increases dramatically in region 2. Since the corona plasma density decreases owing to expansion, the shadow radius expansion rate also decreases. This can be observed from 20 to 40 ps. A smooth shadow radius curve should be obtained,56 indicating an adiabatic expansion process. However, the generation of a converged shock wave and rebounding of the shock wave can introduce high-density plasma into the expansion interface, increasing the corona plasma density, and consequently the shadow radius expansion rate increases again. Therefore, the second increase in shadow radius is a sign of a rebounded shock wave.

In conclusion, the experimental shadowgraph images vividly portray the hallmarks of dynamic converging compression in hydrogen jets, findings corroborated by ray tracing simulations. For a more comprehensive understanding, refining the experimental data is paramount—specifically, minimizing uncertainties, enhancing temporal resolution at around 40 ps to chart the evolution of the reflected shock, and investigations using metal wires of various radii. Leveraging X-ray free electron lasers could furnish direct insights into the shock wave.46 Notably, our analysis presupposes a 1D symmetric compression, a simplification that may not encapsulate the intricacies of the real process. As a consequence, perfect congruence between experimental data and simulations remains elusive.

We have demonstrated a novel method for inducing converging shock compression by utilizing the transient surface return current during femtosecond short-pulse laser–target interactions. The surface return current has been studied by 2D and 3D PIC simulations. With a 5 × 10²⁰ W/cm², 30 fs short-pulse incident laser, the surface return current shows a duration of ∼100 fs, reaching a maximum current of 210 kA. Our analytical approach has examined the impact of the return current on wire targets, focusing on the effects due to the J × B force compression and Joule heating. Intriguingly, the mechanism of convergent shock compression is found to be heavily influenced by the plasma β. Here, magnetic field compression takes the lead when plasma β < 1. Considering the case of a 2.5 µm radius hydrogen jet: the current density confines itself within a skin depth of roughly 0.1 µm, displaying a peak value of 0.28 MA/μm². The hydrogen density is compressed by a factor of 29, resulting in a peak pressure of 100 TPa and a temperature of 220 eV when plasma is imploded in the center. A scaling law has been proposed to extend the implications of the return current to targets of various atomic numbers Z and radii. The results suggest that as the target radius and Z increase, surface ablation becomes the dominant mechanism initiating the convergent shock wave. In the ranges of Z and radius investigated (Z ≤ 50 and r ≤ 20 µm), a shock wave with a velocity U_s > 100 km/s can be initiated in all cases. Utilizing PIC and hydrodynamic simulations, alongside experiments with 2.5 µm radius hydrogen jet and optical shadowgraphy ray tracing simulations, we have observed rapid expansion of hot surface plasma, the presence of cold bulk plasma, and reflective behavior of the shock wave. These findings indirectly confirm our hypothesis of convergent shocks initiated by the surface return current. In summary, our discoveries shed light on the dynamics of return current-initiated converging shock compression during laser–solid interactions. More than just a revelation, our work paves the way for innovative methods to realize high densities and pressures using Joule-level short-pulse lasers.

Acknowledgment. FLASH was developed in part by the DOE NNSA- and DOE Office of Science-supported Flash Center for Computational Science at the University of Chicago and the University of Rochester.