Multi-messenger dynamic imaging of laser-driven shocks in water using a plasma wakefield accelerator

Domain: physics
Published: 2025-12-16
Editorial outcome: minor_revision

Abstract
Understanding dense matter hydrodynamics is critical for predicting plasma behavior in environments relevant to laser-driven inertial confinement fusion. Traditional diagnostic sources face limitations in brightness, spatiotemporal resolution, and in their ability to detect relevant electromagnetic fields. In this work, we present a dual-probe, multi-messenger laser wakefield accelerator platform combining ultrafast X-rays and relativistic electron beams at 1 Hz, to interrogate a free-flowing water target in vacuum, heated by an intense 200 ps laser pulse. This scheme enables high-repetition-rate tracking the evolution of the interaction using both particle types. Betatron X-rays reveal a cylindrically symmetric shock compression morphology assisted by low-density vapor, resembling foam-layer-assisted fusion targets. The synchronized electron beam detects time-evolving electromagnetic fields, uncovering charge separation and ion species differentiation during plasma expansion – phenomena not captured by photons or hydrodynamics simulations. We show that combining both probes provides complementary insights spanning kinetic to hydrodynamic regimes, highlighting the need for hybrid physics models to accurately predict fusion-relevant plasma behavior.

Introduction
Inertial confinement fusion (ICF) recently achieved a historical milestone by reaching a target gain of unity1–5, reigniting optimism for fusion energy as a transformative and sustainable solution. A key challenge in ICF lies in understanding hydrodynamic instabilities during fuel compression, which are critical for the behavior of the burning plasma. To address this challenge, X-ray radiography has long been an essential tool in high-energy-density physics (HEDP)6–8, permitting the diagnosis of plasmas with densities normally exceeding ρ ≥ 1 g cm−3.

High-energy laser facilities commonly irradiate metal, foam, or gas targets with UV laser beams to create X-ray backlighters9–11. While these sources coupled with streak cameras can capture many HED-relevant processes, they suffer from poor brightness due to low conversion efficiency, and limited spatiotemporal resolution typically exceeding Δx > 10 μm and Δt > 200 ps.

In parallel, electric and magnetic fields are increasingly recognized as playing significant roles in ICF environments12. The high temperatures and steep gradients mean that kinetic effects and electromagnetic fields may influence the system dynamics. In this sense, charged particle beam probes have also proven useful in diagnosing these complex interactions13. Although previous studies have proposed simultaneous imaging using different particle types14–16, such as X-rays and protons, they are similarly founded on traditional backlighter-based sources facing many limitations. Some of the challenges include restricted proton energy, spectral modulation, limited simultaneity between the probes, and low temporal resolution. Moreover, these methods typically rely on single-pulse, static configurations, limiting their ability to probe time-dependent dynamic systems. New methods that introduce complementary imaging probes could overcome some of these challenges, offering enhanced spatiotemporal resolution and the integration of field-sensitive diagnostics to study complex plasma interactions.

As a promising approach, laser wakefield acceleration (LWFA)17 has emerged over the past decades as a table-top source of relativistic electrons and X-ray pulses18–24. The betatron X-rays are ultrafast25,26, have a small source size27 enabling high spatial resolution, are brighter than conventional backlighter-based light sources28, and are suitable for high-repetition-rate imaging. Additionally, after propagation of a few cm the photons develop spatial coherence29, making them compatible with phase-contrast X-ray imaging (PCXI)30–33 techniques. Recent work has demonstrated betatron X-ray capabilities in high-resolution imaging34–36, X-ray absorption spectroscopy37,38, among other applications39. Moreover, LWFA electron beams have also been used as probes for laser-matter interactions40–43, aiming to develop an imaging technique that is sensitive to electromagnetic fields and has ultrafast timing resolution. However, previous LWFA studies have been limited exclusively to either a photon or electron probe, without integrating both to achieve a deeper understanding of the system in question.

In this work, we show that by utilizing both high-resolution X-ray photons and coordinated, electromagnetic field-sensitive electrons, one can reveal experimental details that would remain hidden from using a single particle type alone. In a “multi-messenger”-like approach akin to astronomy, one obtains insights about the whole of the interaction, revealing a picture that is greater than the sum of its parts44. Moreover, previous LWFA experiments have been restricted to single-shot, solid targets, and many have not been able to exploit the potential of high-repetition-rate laser capabilities in combination with liquid targets to image time-dependent dynamic systems.

In this study we capture the full dynamic evolution of a laser-heated ablating plasma and shock-compressed water column using pump-probe imaging of the interaction with both X-ray and charged-particle beams. X-rays capture the hydrodynamic shock development, while the relativistic electron beam measures the generated electromagnetic fields in a non-invasive radiographic scheme. Importantly, both particle sources are correlated in size and synchronized in time. While X-ray or electron beams alone provide an incomplete picture of the interaction, their combination offers a powerful and complementary tool. This approach reveals that detailed knowledge of target conditions along with a holistic view on laser-plasma evolution are needed for accurately modeling the complex plasma dynamics.

Results
The experiments were performed at the Lawrence Berkeley National Laboratory BELLA Center, where the Hundred Terawatt Thomson (HTT) dual beam system was used to drive both the laser wakefield accelerator and the laser target ablator, as shown in Fig. 1. In combination, a free-flowing liquid (water) target was designed that is capable of providing a bulk plasma (30 μm diameter stream) in vacuum. The custom-made water jet (“Water target”) is replenishable, clean, and suitable for laser-driven high-repetition-rate experiments. The HTT laser system (λ = 800 nm, 1 Hz repetition rate) consists of a main LWFA beamline with a linearly-polarized (1.5 ± 0.2 J), ultrafast (pulse duration 40 ± 5 fs FWHM, peak power 33 ± 6 TW) laser pulse focused by an f/20 off-axis parabola to a 20 ± 4 μm FWHM spot, incident on a gas target (“Gas target”) reaching on-target peak intensities of I = [7 ± 2] × 1018 W cm−2. A second shock-driver laser was obtained by splitting the main BELLA HTT beam and subsequently bypassing the compressor, thus generating a long high energy pulse (200 ps FWHM, 1.0 ± 0.2 J). The secondary pulse was focused on a spot of size 20 ± 5 μm along the x direction, perpendicular to the main beam axis z, and reaching intensities of I = [1.4 ± 0.2] × 1014 W cm−2. A variable delay stage on one of the beamlines allowed for precise control of timing between the shock-driver pulse with respect to the LWFA probes up to a maximum of 8 ns within the interaction. Laser pointing fluctuations of ± 15 μm RMS were mitigated by active feedback correction and by setting the focus of the long pulse laser past the water jet, thus employing a spot size larger than the target diameter (w ~ 60 μm) at its plane.

Once the short laser pulse travels through the gas jet the ponderomotive force expels electrons away from high-field intensity regions leaving ion cavities and launching plasma waves on its wake. Due to the high-field gradients following the pulse, trapped electrons can be accelerated to relativistic velocities during the interaction. The oscillation of these electrons inside the ion bubbles in turn generates betatron X-ray pulses of ultrafast duration. In this sense, a laser wakefield accelerator generates synchronized X-rays and relativistic electrons, providing a unique tool for imaging unlike others where only one type of particle is available. In this study, the electron beam probe, having an average energy of 146 ± 7 MeV, was characterized using scintillating screens and a calibrated dipole magnet downstream (“Electron beam characterization”). The betatron X-rays were recorded using a cooled, in-vacuum, CCD camera (PI-MTE). The X-ray beam spectrum was characterized using a Ross pair filter wheel, resulting in Ec = 4.4 ± 0.7 keV, and a source size in the order of ~1 μm from the diffraction pattern of a sharp knife-edge (“X-ray beam characterization”). The spectra recorded for both probes are displayed in Fig. 1, and the experimental geometry chosen for multi-messenger imaging is further discussed in (“X-ray beam imaging”) and (“Electron beam imaging”).

Initially, when the long pulse laser interacts with the water, it deposits its energy over some range of densities below critical density, in the corona of the pre-formed plasma. This occurs mostly through inverse bremsstrahlung, but in addition through potential hot electron generation mechanisms45. Early preheating of the interior may also occur via laser shine-through effects46. Following laser energy deposition, electrons carry the heat to higher densities above critical density. Consequently, the dense liquid surface is quickly heated, leading to its ablation and subsequent expansion into a vacuum. In response to the large pressures generated at the ablation surface from the expansion of the material, a hydrodynamic shock wave is launched into the liquid target.

For low-Z laser-irradiated targets, reduced opacity minimizes the role of thermal radiation emission in the dynamics, and the low atomic number significantly reduces bremsstrahlung background from electron-target interactions compared to high-Z materials. However, in liquid water (ρ = 1 g cm−3), partial ionization increases opacity, making radiative losses more significant. At these ICF-relevant densities the evolution of the target is complex, involving hot-electron production, electron heat transport, radiation processes, shock-wave generation, and hydrodynamic expansion—all occurring within fractions of a nanosecond under typical experimental conditions. Nanosecond timescales fall within a regime where the complete physics of the interaction is hardly accessible to any single contemporary simulation. Our setup is then ideally suited for experimentally studying such complex dynamic systems, offering ultrafast and sub-micron resolution, nanosecond delay range, and dual probes sensitive to variation in both density and electromagnetic fields.

In this study, the evolution of the laser-water interaction was predicted using FLASH47, a radiation hydrodynamic simulation code (“FLASH simulations”). Projected density maps obtained from FLASH were post-processed with a Fresnel-Kirchhoff-based algorithm (“Fresnel-Kirchhoff algorithm”) to produce synthetic phase-contrast X-ray patterns as shown in Fig. 2. The first main result, displayed in Fig. 3 presents a time-series comparison between synthetically generated phase-contrast X-ray images and experimentally obtained betatron X-ray images, after preprocessing (“Fourier mask”). The laser-driven shock displays very similar features in both simulation and experiment (see Supplemenary Movie 1 and Supplementary Movie 2). The features observed in Fig. 3a include bright phase-contrast enhancement at the edges of the target, as well as a strong, dark, bow-shaped shock structure that grow in y-direction and propagates forward in x-direction. Smaller precursor signals ahead of the main shock are also appreciable and are examined later.

To analyze these results, average lineouts were taken at the center of each image in the time series. The lineouts were then stacked together to create a composite image of the full interaction, shown in Fig. 3b, c. These aggregate images reveal the main shock breakout through the rear of the target around Δt ≅ 2 ns. The lineouts were subsequently analyzed to compare the shock velocity between simulation and experiment, which are comparable in magnitude as demonstrated in the analysis of Fig. 4.

Measuring shock velocity is particularly important because it can provide information regarding the thermodynamic state of the material. By taking the frame of reference where the shock is at rest, one can define an upstream shock Mach number, Mu = us ρ1 / (γp1). Here us represents the shock velocity, ρ1 denotes the unshocked fluid density, p1 is the corresponding initial pressure, and the polytropic index value is assumed to be γ = 5/3. Such index is typical for fully ionized, weakly coupled HED systems where radiative effects are minimal48. For the present experimental parameters Mu ≈ 104, which is a clear indication of the strong-shock regime.

In order to calculate useful parameters for ICF environments, one can utilize the well-known Rankine-Hugoniot jump conditions and obtain predictions for the post-shock density and pressure in terms of Mu as,

ρ2 / ρ1 = Mu2 (γ + 1) / (Mu2 (γ - 1) + 2), (1)

and

p2 / p1 = (2γMu2 - (γ - 1)) / (γ + 1). (2)

The uncertainty in Mu can be substantial, however note that in the limit when Mu ≫ 1, the physical limit for density jump in a polytropic gas approaches ρ2 / ρ1 = (γ + 1) / (γ - 1) ≈ 4, while the pressure grows infinitely, and is strongly dependent on the shock velocity measured experimentally. For the analysis shown in Fig. 4, utilizing us ≈ 20 μm/ns, the resulting post-shock pressure on target is p2 = 3–4 Mbar, comparable in magnitude to that at the Earth’s core. This pressure scales with laser intensity as p2 ∝ 6.1 I14 2/3 λu 2/3 Mbar. Where I14 is the intensity in units of 1014 W cm−2 and λu is the laser wavelength in micrometers48.

Moreover, one can utilize the calculated pressure p2 and the measured shock velocity us to obtain a useful prediction for the ion temperature. By taking p2 = (Z2 + 1) kB Ti,2 ρ2 / (Am p) and taking the strong shock limit one finds

kB Ti,2 = (A + Z2) / (A + 1) * (p2 / ρ2) * ((γ - 1) / 2). (3)

Assuming the electrons are non-degenerate, they fully equilibrate with the ions, and ignoring Coulomb modifications to the pressure, the immediate post shock temperature of the ions before they equilibrate with the electrons can be found by setting Z2 = 0, corresponding to the average ionization state of the post-shock ions. Then Eq. (3) results in an ion temperature of Ti,2 = 4.7 eV.

Notably, although early time agreement with simulations was good regarding the shock propagation velocity, the simulated shock structures did not fully match the experimental measurements at intermediate times. In Fig. 5a, panel (1) represents a simple 3D (or standard 2D) FLASH simulation of a cylindrical water target heated by a laser pulse. Interestingly, ten-shot averaged panel (3) and single-shot panel (4) show X-ray images of the shocked target at near maximum compression at Δt ~ 1.0 ns. These display an apparent rear-driven shock structure that is different from conventional 2D, and simple 3D, FLASH simulations.

Analysis of the plasma conditions indicates that it is unlikely to observe a strong shock reflection at the rear interface. Instead, a much better agreement with the structure produced experimentally was achieved in Fig. 5a panel (2), by incorporating a low-density layer surrounding the target, approximating the vapor expected from water evaporation in vacuum. Including the low-density layer in the simulations enhances thermal transport around the water column, resulting in more uniform heating of the jet’s exterior and producing a cylindrical shock49 and symmetric compression morphology closely matching the experimental observations.

To illustrate this effect, Fig. 5b displays 2D slices from 3D simulations of the density and pressure profiles taken at the midpoint, x–z plane, of the water cylinder. Panel (1) shows a simple water target without surrounding vapor, where the laser-driven shock is strong and predominantly one-sided. In contrast, if a vapor density profile falling as ∝ 1/r from the surface is introduced in panel (2) a key difference in compression morphology is observed. The partially ionized vapor blanket provides a low-opacity, low-density medium in which electrons are able to transport energy more efficiently around the surface of the target. This leads to more uniform density and pressure profiles, and to the generation of a cylindrically symmetric shock structure.

To obtain a deeper understanding of this phenomenon, consider that the laser energy at the peak of the pulse is deposited in ionized water electrons below the critical density nc. This creates a high-temperature plasma region where electrons ~ 1 keV are able to escape from the electrostatic forces imposed by the ions, but are free to move around the surface of the target traveling a significant distance before depositing their energy. For instance, let us compare the mean-free-path of a 1 keV thermal electron carrying heat in the plasma region below n ≤ nc to that of an electron in the dense solid n ≈ ne0. For the low density region with nc ~ 1021 cm−3 we obtain λmfp ~ 20 μm, in the order of the diameter of the stream. By contrast, for the high-density region n ~ 1023 cm−3 the mean-free-path is λmfp ~ 0.2 μm. As a result, the heat flow within the target would be localized, while thermal transport in the vapor layer would be highly nonlocal.

The experimental observations in Fig. 5a panel (3) and panel (4) have led us to better understand the compression morphology of the target and the importance of accurately modeling the experimental conditions. To support this interpretation with observational evidence, we now make use of the relativistic electron beam probe generated by the laser wakefield accelerator. The nonlocality of the heat transport and the presence of energetic electrons around the interaction region, result in the production of electric (through the strong pressure gradients) and magnetic (through current flows) fields in the plasma surrounding the water target. These field-generating mechanisms are well known but are often not included in fluid-based hydrodynamic codes, as electric fields are typically precluded by enforced quasineutrality. Unlike the X-ray probe, the LWFA electron beam will be sensitive to path-integrated deflections caused by electric and magnetic fields along its trajectory. These perturbations are captured downstream on the profile imager, forming an image of the interaction.

Using this approach, the electron beam probe was used to capture the evolution between the long pulse laser and the water stream, as shown in Fig. 7a (see Supplementary Movie 3). Bright features displayed on the scintillator screen indicate accumulation of probe electrons from focusing fields, while dark features would typically represent absence of electrons from defocusing fields. The first noticeable feature in the time-series is a bright channel in the laser propagation direction, which persists for approximately the pulse duration of 200–300 ps and focuses the electron probe. The fields are likely to be electric, arising from the pressure gradient caused by the long pulse laser ionization and heating. Notably, the observation of such early-time ionization channel across both sides of the water jet further supports the idea of nonlocal heat transport, hot electron generation, and a symmetrically ablated target observed with the X-rays in Fig. 5. Although the channel forming on both sides of the liquid jet might suggest laser transmission through the water, it is important to recall that the long-pulse laser beam is larger than the water column diameter, allowing a significant portion of the laser energy to bypass the target.

The second prominent feature observed with the electrons is a dark plasma cloud expanding from the center, indicating the presence of complex, strong electromagnetic fields. While this dark structure might initially be interpreted as defocusing fields, the absence of a plausible physical mechanism from simulations in Fig. 8 suggests the fields are most likely overfocusing the probe. Moreover, the sharp bright circular ring surrounding the plasma cloud indicates the presence of caustics in the image. Although caustics complicate the quantitative analysis of the fields, lower bounds on their strengths and key features of their topology may still be extracted using a field recovery method (“Field recovery method”). From Fig. 7c, we infer integrated fields on the order of ∫Edl ~ 104 V or ∫Bdl ~ 10−4 T·m. In the case of electric fields, a simple estimate using E ~ ∇P/ene ~ ∇(kBTe/e) would imply hot electrons with temperatures on the order of keV.

More importantly, a time-dependent morphological understanding of the fields is acquired from Fig. 7, where different plasma species can be identified from the radiographic images. First, there is a dark inner cloud bounded by a bright ring, likely caused by strong overfocusing fields from an oxygen ion plasma. Second, a fainter dark outer ring, which may be attributed to a hydrogen ion (protons) plasma marking the boundary of its expansion into vacuum. At the plasma edge, a sheath field forms due to the separation of electrons and protons by an amount of around the local Debye length. This sheath field balances the thermal motion and exerts a positive force on the electrons in the negative radial direction, focusing (or overfocusing) the probe. The timescale of expansion indicates that the dark outer ring must originate from an expanding proton plasma defocusing the probe, as its expansion is too slow to be attributed exclusively to defocusing electrons given the working temperatures.

Within this framework, the expansion velocity analysis in Fig. 9 presents both a slower oxygen plasma (O+, inner cloud) with average atomic number Z = 8 and velocity of uO = 191 ± 21 μm/ns, as well as a second hydrogen plasma species (H+, outer dark ring) with a much faster characteristic expansion speed of uH = 731 ± 39 μm/ns and much lower average atomic number of Z = 1. The speed of this plasma boundary is consistent with protons moving with the same average energy as the slower expanding oxygen ions. Protons are expected to be accelerated from the surface of a hot plasma51, however, this feature is not present in the single-fluid simulations. To approximate a water molecule, the FLASH plasma fields use a single mean atomic number of Z = 3.33 expanding at a velocity of uF = 260 μm/ns, and falling between the recorded oxygen and hydrogen curves.

Hence, the relativistic electron beam probe is able to provide evidence for an expanding hot plasma surrounding the water column not visible by the X-rays, as well as to capture the evolution of different ion species not present in the simulation. More importantly, the dual-sided near-isotropic ablating plasma, observed with the electron beam probe, would support the model of a cylindrically symmetric compression shock morphology observed with the X-rays.

The absence of hot electron populations in FLASH simulations is particularly relevant in light of our observations. Hot electrons generated during laser-plasma interactions can travel significant distances, driving non-local heat transport and depositing energy into the surrounding vapor layer. These findings underscore the need for advanced diagnostic systems in future HED experiments, as well as hybrid simulations that incorporate kinetic effects alongside radiation hydrodynamics.

Methods
X-ray beam imaging
The experimental geometry was carefully designed to enable a propagation-based phase-contrast X-ray imaging configuration. To properly choose a source-to-object distance R0 and object-to-detector distance R1, the coherence length L⊥ = R0 / kσ was estimated for a collection of uncorrelated emitters of wavenumber k and source size σ27. To resolve features of order σ at the image plane, the coherence length must exceed the source size (L⊥ > σ). Rearranging, the source-to-object distance can be estimated as R0 > kσ2, which in terms of the photon energy in units of keV and microns is

R0 [m] > 5.1 × 10−3 (σ [μm])2 (ℏω [keV]). (4)

For 5 keV X-rays and a σ = 2 μm source size, R0 > 10 cm ensures that the coherence length is larger than the source size.

The broadband polychromatic spectrum of a betatron source supports propagation-based phase contrast since, to first order in the paraxial approximation, the phase-contrast pattern is independent of wavelength when the effective Fresnel number is larger than unity (NF,eff > 1), where

NF,eff = (R0 λ)−1 * (M2 σobj2 + (M-1)2 σ2 + PSF2) / (M-1), (5)

M is the magnification, σobj is the size of the smallest object feature, and PSF represents the detector’s point-spread function as described in ref. 54. Therefore, the condition NF,eff > 1 holds true given our experimental setup configuration with source-to-object distance R0 = 16 cm, object-to-detector distance R1 = 580 cm, and beam divergence of a few tens of mrad.

X-ray beam characterization
The X-ray radiation spectrum was characterized using a Ross filter wheel, as described in ref. 55. The filter wheel consisted of wedges made of different materials and thicknesses to selectively attenuate parts of the spectrum. By analyzing the transmitted intensity map and fitting a synchrotron-like spectrum56, the critical energy of the X-ray beam was determined to be Ecrit = 4.4 ± 0.7 keV.

The size of the X-ray source was estimated using a sharp “knife-edge” placed in the beam path. The intensity profile of the transmitted beam was measured and fitted to the expected Fresnel diffraction pattern produced by a Gaussian source interacting with a half-plane. This analysis yielded an upper bound for the X-ray source size of σ ≤ 1 μm.

To obtain a nominal background signal, we recorded images without gas or water flow and averaged over 10 shots to improve the signal-to-noise ratio. The background was then subtracted from each signal image on a shot-by-shot basis. To protect the in-vacuum CCD camera and further reduce background signals, a filter assembly was placed upstream from the detector. This consisted of a 60 μm aluminum foil, a 25 μm kapton film, and a 36 μm mylar layer, which together served to attenuate residual laser light, suppress plasma self-emission, and shield the detector from debris.

Electron beam imaging
The geometry for electron-beam imaging shared the same source-to-object distance as the X-ray imaging setup, R*0 = R0 but used a significantly shorter object-to-detector distance R*1 = 141 cm. This configuration was selected to capture the full electron beam profile on a phosphor scintillating screen located downstream in the chamber, prior to the dipole magnet. This imaging configuration provided a radiographic picture of the electron beam profile and its perturbation from the interaction fields.

By employing distinct object-to-detector distances for the X-ray and electron-beam imaging setups, this configuration provides a dual perspective of the interaction with different magnifications. The electron radiography setup allowed for a broader field of view of the beam profile, complementing the higher-resolution perspective offered by the X-ray imaging.

Electron beam characterization
The relativistic electron beam produced by the wakefield accelerator was characterized using a magnetic spectrometer57, comprising a 1.5–3 T electromagnet and a 1 T permanent dipole magnet along the beamline. The magnetic fields dispersed the electrons based on their momenta, projecting their trajectories onto a series of Lanex scintillating screens, from which the electron-induced fluorescence was imaged onto an array of 12-bit CCD cameras.

The beam was characterized by mapping the electron positions on the screens to particle tracking simulations through the experimentally measured magnetic field. This process allows for precise beam energy, divergence, and charge calibration as described in ref. 58.

The mean energy of the electron beam for the experiment was 146 ± 7 MeV as shown in Fig. 1, with a pointing divergence of 7.8 ± 1 mrad in the x-direction and 3.4 ± 0.8 mrad in the y-direction. The mean charge of the beam was 24 ± 4 pC. For the analysis of field recovery from electron radiographs, a representative beam energy of E0 = 44 MeV was selected. This lower energy reflects operating conditions where the beam was intentionally detuned to increase divergence and maximize field sensitivity across the imaging region. Additionally, this choice allows chromatic deflection effects to be better accounted for when measuring field magnitudes.

Gas target
The laser plasma accelerator employed ionization injection59 to generate a relativistic electron beam by focusing the high-power laser pulse into a 3 mm mixed gas jet60 composed of 99.5% helium and 0.5% nitrogen. The supersonic gas jet utilized a fast solenoid valve (Parker Pulse Valve) synchronized and triggered alongside the high-power beamline. A voltage-controlled regulator allowed precise tuning of the gas jet’s backing pressure, enabling adjustment of the gas plume density within the range n ∈ [2.1, 2.8] × 1018 cm−3. To characterize the laser-plasma interaction, a Mach-Zehnder interferometer was utilized, incorporating a frequency-doubled probe beam oriented perpendicular to both the gas jet and the high-power laser line. This configuration enabled synchronized diagnostics of the plasma density profile61.

Water target
A custom-made cylindrical liquid water jet flowing in vacuum was employed as the target for the experiment, drawing inspiration from existing systems such as bulk liquid targets used in time-of-flight mass spectrometers62 and thin liquid sheets63.

The water was delivered to the chamber by a high-performance liquid chromatography (HPLC) pump connected to 1/16” OD stainless steel tubing (0.03” ID). The jet nozzle consisted of cleaved Polymicro (Molex) capillary tubing with 30 μm ID and 360 μm OD. The water was delivered at a flow rate kept between 1–2 mL min−1, maintained constant by the HPLC pump.

The water collector assembly included a conical copper head with a top aperture of d ≈ 500 μm, designed to capture the water stream. To prevent ice formation during the alignment procedure, the collector head was heated by embedded miniature cartridge heaters (Thorlabs, 15W). The collector assembly is connected to a reservoir via standard vacuum components.

The alignment of the water jet stream with the collector was achieved using picomotors for small adjustments along the x, y, and z directions. Two orthogonal cameras positioned outside the vacuum chamber monitored the procedure until stable operation was achieved after pump-down.

After alignment, chamber pressures as low as 10−5 Torr could be achieved using >1000 L s−1 turbo pumps and a liquid nitrogen cold trap. The stable water jet was subsequently aligned with the long pulse laser beam using translation stages supporting the whole assembly.

FLASH simulations
Three-dimensional (3D) radiation-hydrodynamic simulations were performed using FLASH47 to predict the evolution of the laser-plasma interaction. FLASH is a multi-dimensional, radiation hydrodynamic simulation code based on the Eulerian approach. It features dynamic block adaptive meshing, treats multiple materials, includes electron heat conduction and related physics, transports radiation via multi-group diffusion, and deposits laser energy by tracing rays in 3D. Although FLASH simulations accurately model the fluid dynamics, they do not include electric (E) and magnetic (B) fields nor treat hot electron populations.

The simulations utilized a 3D Cartesian coordinate system consisting of a cylindrical plasma target with a radius of r0 = 15 μm and mass density ρ0 = 1 g cm−3. A surrounding chamber plasma was initialized with a much lower initial mass density ρc. Two different density profiles for the chamber plasma were explored: 1) a standard target with constant ρc = 10−6 g cm−3 throughout the domain, and 2) an evaporative target so that ρc = ρ0 (r0 / r2) with ρ0 = 10−3 g cm−3 following a decaying distribution as r increases.

Both the target and chamber plasmas were modeled with the same effective atomic number Zeff,t = Zeff,c = 3.33 and average atomic mass Aeff,t = Aeff,c = 6.0. The plasmas were similarly initialized at room temperature Tt,0 = Tc,0 = 290 K, and the multi-group flux-limited diffusion coefficient was chosen to be f = 0.17, considering our laser parameters and ref. 64. The equation of state for both plasmas was calculated using the PrOpac EOS code.

The shock-driver laser pulse in the simulations was modeled with λ = 0.8 μm using a truncated Gaussian profile for its intensity with FWHM = 40 μm. The laser power deposited into each cell is calculated based on inverse bremsstrahlung, which depends on the local electron number density gradients and the local temperature gradients. The laser pulse was defined in sections using a piecewise linear function, where each section is associated with a time-power pair. These pairs ensure that the total energy in the pulse (1 J) is delivered over the total time window (220 ps).

Image processing
Fourier mask. We applied a custom filter in the Fourier domain to process the experimental X-ray data, reducing undesired high spatial frequencies and thereby increasing the signal-to-noise ratio. The method involves constructing a mask that selectively filters spatial frequencies based on a combination of radial and elliptical criteria.

Given an input image I(x, y) of dimensions Nx × Ny, the two-dimensional Fourier transform Ĩ(kx, ky) = F{I(x, y)} is computed first, where kx and ky represent the spatial frequency components in the x and y directions, respectively.

Next, a radial frequency component K = (kx2 + ky2)1/2 and an elliptical frequency component Kel = (a2kx2 + b2ky2)1/2 are defined. The final frequency-space mask M(kx, ky) is then constructed by incorporating these two terms as:

M(kx, ky) = (1 - exp(-K/Kmax)) * |sin(π Kel / Kmax)|N (6)

where Kmax = π is the maximum spatial frequency, and N is a parameter that controls the sharpness of the mask. To avoid singularities when K = 0, we set the corresponding values in the mask to 1. Moreover, we apply a frequency-space mask to the image as Ĩmasked(kx, ky) = Ĩ(kx, ky) * M(kx, ky), and the inverse Fourier transform is used to recover the masked image in real space:

Imasked(x, y) = F−1{Ĩmasked(kx, ky)} (7)

Fresnel-Kirchhoff algorithm. To model the expected image pattern from a phase-contrast imaging system, we follow a common approach outlined by Born and Wolf65 and applied in past work66,67, which consists of solving the Fresnel-Kirchhoff integrals numerically to obtain the expected complex wave-field distribution ue after propagation from an initial point at the source, to a point P = (x, y) in the detector plane.

In this context, the real “pure” pattern I is given by

I = I0 |ue(P)|2, (8)

where I0 is the incoming intensity distribution at the sample. Assuming that the effective Fresnel number is larger than unity, the complex-valued wave-field distribution can be simplified utilizing the paraxial approximation. This approximation yields a final expression for the complex-valued wave-field distribution in Fourier space as given by ref. 54,

Ue(u, v) = M2 T(Mu, Mv) * exp[-πiλR1 M(u2 + v2)] * exp[2πi(R1/R0)(xu + yv)], (9)

where u and v are the transverse spatial frequencies corresponding to x and y coordinates, respectively. Equation (9) can be solved using a fast Fourier transform (FFT) algorithm to obtain the pure pattern I, which can then be convolved with the source size σ to obtain the real phase-contrast pattern in the detector. Here T(Mu, Mv) is the Fourier transform of the object transfer function t(P) = exp[iϕ], where the phase induced ϕ is complex-valued and depends on the material properties as

t(P) = exp[i(D(P) - B(P))] (10)

For X-rays crossing a sample, the index of refraction is less than unity and has the form n = 1 - δ - iβ. The phase map is then calculated using Eq. (10) and the following projected distributions obtained from radiation hydrodynamics simulations,

D(P) = (-2π / λ) ∫ δ(x, y, z) dr (11)

and

B(P) = (-2π / λ) ∫ β(x, y, z) dr (12)

where δ and β are the real and imaginary components of the refractive index and are expressed as,

δ = re NA λ2 ρ / A * Σj [Zj + f0j] (13)

and,

β = (λ / 4π) * (ωp / c)2 * (ν / ω) (14)

where re is the classical electron radius, NA is the Avogadro number, ρ is the mass density, Aj are the atomic number and the atomic weight of the j-th element of molecule, f0j is the real part of the dispersion factor, ωp is the density-dependent plasma frequency, and ν is the collision rate factor.

Finally, to account for the polychromaticity of the source, the resulting image can be weighted and summed according to the spectrum distribution of the source

Ipoly = Σ w(E) Imono(λ), (15)

where w(E) is the energy-dependent weighting factor obtained from the source spectrum.

Meijering filter. The Meijering filter is an image processing technique to accurately quantify and segment neurite-like traces in fluorescence microscopy images, as described by Meijering et al.68. In this work the algorithm is adapted for the detection of shocked traces in the sample, which deviate from the nominal un-shocked image.

For the implementation of the algorithm, the modified second-order derivative of the image at position x = (x, y) can be computed as,

fij(x) = (f * Gij)(x) (16)

where

Gij(x) = ∂2 / ∂i ∂j G(x) (17)

and the derivative directions i and j can be either x or y.

Next, the eigenvectors and eigenvalues are computed in the Meijering algorithm by using a modified second-derivative matrix given by,

H0f(x) = [fxx(x) + α fyy(x) fxy(x); fxy(x) fyy(x) + α fxx(x)] (18)

where α is a parameter which needs to be optimized. The normalized eigenvectors vi(x) and their corresponding eigenvalues λi(x) in the standard Hessian second-derivative matrix are then obtained as,

v01(x) = v1(x)
v02(x) = v2(x) (19)

and

λ01(x) = λ1(x) + αλ2(x)
λ02(x) = λ2(x) + αλ1(x) (20)

In this way, the algorithm calculates the eigenvectors of the Hessian to determine the similarity of an image region to the shocked traces in the sample. The Meijering filter was applied to both experimental and simulation X-ray images using its default parameters as given by scikit-image filters package.

Field recovery method. Following the technique described by Kugland et al.69, the field recovery method begins by assuming that the electron beam probe, with object-to-detector distance R1 and source-to-object distance R0, is deflected by electromagnetic fields present at the interaction region. Generally, R1 ≫ R0, and R1 ≫ a, where a is a characteristic spatial scale of the field in question. In a Cartesian geometry, the ideal image I0 at the object plane for an undisturbed electron beam is described by the coordinates (x0, y0), while the real image I in the detector is described by the coordinates (x, y), such that x = x0 * M and y = y0 * M, where M = (1 + R1/R0) is the geometric magnification.

After the electrons traverse the electromagnetic fields, they are deflected by angles αx and αy. The coordinates of the real image, as described by the electrons’ trajectories, can be approximated, when αx,y are small, by

x = x0 + R1 x0 + αx R1, (21)
y = y0 + R1 y0 + αy R1, (22)

Therefore, the objective of the algorithm is to determine the deflection angles αx and αy, as they can be related to the path-integrated electric or magnetic fields present at the interaction region by the following expressions,

αx = (q / γme v2) ∫ Ex dz
αx = (q / γme v) ∫ By dz,

and

αy = (q / γme v2) ∫ Ey dz
αy = (q / γme v) ∫ Bx dz (23)

Finally, the disturbed beam at the detector plane I(x, y) can be then obtained by the following relation,

I(x, y) = I0 / |∂(x, y) / ∂(x0, y0)|, (25)

Here, |∂(x, y) / ∂(x0, y0)| is the absolute value of the determinant of the Jacobian matrix that relates the object and image planes and can be described by

|∂(x, y) / ∂(x0, y0)| = (1 + L1 + ∂αx/x) * (1 + L1 + ∂αy/y) - L2 ∂αy/x ∂x αy (26)

To solve this equation, we assume that α = [αx, αy, 0] is small and that the object-to-image coordinates follow a linear mapping. Additionally, all electrons’ trajectories are assumed to be rotation-less such that ∇×α = 0. For this regime, the deflection angles can be obtained from a potential field ϕ so that

α = ∇Φ: (27)

For convenience, we introduce the following normalization: xe = x/wx, ey = y/wy, αei = αi L/wi, αe = ∇eΦe, and Ie0 = I0/M2 where wi is width of the beam at the image plane. The Jacobian, described by Kugland et al., can then be rewritten as,

(1/M2) |∂(x, y) / ∂(x0, y0)| = (1 + ∇e2Φe + ∂e2Φe/∂xe2 ∂e2Φe/∂ey2 - (∂e2Φe/∂xe∂ey)2) (28)

This leads to the final equation, which is solved numerically,

∇e2Φe = (I0 / I) - 1 - (∂e2Φe/∂xe2 ∂e2Φe/∂ey2 - (∂e2Φe/∂xe∂ey)2) (29)

Under strong-field conditions, or in the presence of low-energy electrons, large deflection angles αx,y can lead to overlapping beam trajectories, and the formation of caustics in the radiograph. This affects the one-to-one mapping between the source and detector coordinates, making the relationship between x and x0 nonlinear and the correspondence between I and I0 non-unique. Moreover the broad energy spectrum of the electron beam can further introduce chromatic effects in the deflection response, these should not be large as described in ref. 42. Nevertheless, key features of the field topology, its evolution in time, and approximate lower bounds on field strength may still be robustly extracted using this method.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper reports on laser-driven shocks in water, observed through secondary sources generated by a laser plasma accelerator.\nFirstly, I find this work highly impressive and original. The quality of the data is very good, and the choice of diagnostics is highly relevant. It is particularly exciting to see shock propagation characterized at such a high repetition rate. This achievement is the result of the careful selection of the laser system, target, and X-ray betatron and electron beams, which together made this work possible.\nThe article presents interesting experimental data and numerical results of high quality. While the suggested explanations appear plausible, additional analysis and numerical simulations would further strengthen the conclusions. The following comments need to be addressed for the paper to be accepted.\n1. The data related to shock propagation as shown in Figure 2 from the X-ray images indicated a prompt “compression”: What is the origin of the signals that appear at all times (except at 0 ns) ahead to the right (and sometimes to the left) of the shock?\n2. Since the relevant data exist, it would be beneficial to include the corresponding numerical data in Figure 2c. This addition would help track the comparison from t = 0 up to a few nanoseconds. As it currently stands, there is a gap in the information between 0.83 ns and about 8 ns (with only one time step at 4.5 ns).\n3. Why is the ablated plasma not visible at all? One would expect to observe a plasma plume, at least during the duration of the heating laser pulse.\n4. The linear scale in Figure 4b should be replaced with a logarithmic scale (as done in Figure 9 for the temperature). This change would facilitate a better understanding of plasma expansion and ablation.\n5. The electron distribution appears quite surprising. What is the origin of the two main peaks at around 45 MeV and 170 MeV? A continuous injection and broad spectrum would typically be expected. This anomaly should be clarified and explained through numerical simulations and/or by considering instrumental responses.\n6. The unit for the electron energy distribution could be changed to pC/MeV.\n7. The analogy with low-density foam is very interesting. However, for it to be fully relevant, a comparison between the density of the foam used in fusion experiments and the vapor density employed in the FLASH simulations would be necessary. It appears that the vapor density used in FLASH simulations may be of several orders of magnitude lower than that of the foam.\n8. The explanation of the role of different plasma species could be strengthened by including FLASH simulation results showing the distributions of electrons, protons, and oxygen species.\n9. Figure 9 shows sharp gradients and unusual features, such as the rectangular shape in the first image extending from in microns -250 to 0 along the x-axis and from -100 to 100 along the y-axis. Other sharp gradients visible in the images should also be discussed.\n10. Why justify the choice of the flux limited diffusion coefficient to 0.06, and how the results change with other values?", "reviewer_2": "The manuscript presents results of a pump-probe experiment where relativistic electrons and betatron X-rays produced with a laser-plasma accelerator are used to probe the dynamics of a water sample heated by a secondary laser pulse.\nThe X-rays provide a radiograph of the shock front. In parallel, the relativistic electron beam is used to qualitatively track electromagnetic fields in the plasma formed by the laser-water interaction.\nThe main achievements or points of novelty are:\n- this is a real pump-probe application, while many recent articles using betatron as a probe have been limited to \"cold\" samples, without any dynamical study.\n- the application to shock propagation imaging. Up to now, the use of betatron X-rays for this application was restricted mainly to the article of Wood et al. Scientific Reports 2018. Here, there is moreover a deeper analysis of plasma conditions and the results are supported by hydrodynamic simulations.\n- The use of liquid water as a sample, which has an interest for some range of high-energy density experiments. It is also technically appealing since it allows for continuous renewal of the target\n- The use, in combination to X-ray radiography, of the electron beam to probe the plasma.\nThe analysis, though it includes important steps of image processing and data filtering, seems consistent. I am however not a specialist of laser-generated shocks in matter, and I hope that other reviewers are more experts of this physics.\nIn my opinion, the work will be of significance to the field and deserves publication in a high impact journal.\nHowever, I have some concerns and questions:\n- while reading the manuscript, I had sometimes the feeling that the Authors try to oversell the results - whereas I do not think it is useful. The repetition and highlights all along the text of the \"multi-messenger approach\", and some specific sentences such as \"The simultaneous inclusion of ultrafast X-rays and an electric/magnetic field-sensitive probe in LWFA introduce a novel multi-messenger imaging technique\" in the introduction, imply a technological breakthrough and/or a novel scientific approach, which is not really the case. The fact that the e-beam probe can complement the X-ray imaging is indeed interesting, but it is not a paradigmatic change. Moreover, the experimental layout relies on placing the sample before the dipole magnet instead of placing it after, which is not very complicated (and is usually done to avoid perturbation of the sample itself or noise in the X-ray detectors). I also guess that such layout was driven by geometrical considerations for phase-contrast imaging criteria and magnification considerations. I suggest that the Authors \"de-emphasize\" the message on the multi-messenger approach.\n- In the same vein, I suggest to remove the first sentence of the abstract, because this is not (and far from) an ICF experiment, and it is probably not the main motivation at this point. Nevertheless, I completely agree with the last two sentences of the conclusion, about the perspectives of this work in the field of ICF, HED science and for the development of such sources and diagnostics at large-scale fusion facilities.\n- there is no discussion about the repeatability and data statistics. Were all the measurements taken in a single shot (for a given delay and a given probe source)? What about the repeatability of the measurements ? Since the sample is fluid, isn't it difficult to reproduce shot-to-shot the same conditions? (shock front position, EM-fields, etc.). It would be interesting to see a video or a bunch of pictures of the shock images, taken at the same laser and delay conditions. The stability of the observed phenomenon is crucial here, especially as the two probes do not record the same shots.\n- Was there a background substration on the X-ray imaging measurements? If yes, was it taken with or without water flow, with or without accumulation on several shots?\n- Usually, phase-contrast effects can be clearly seen on lineout profiles in the radiograph. Could the authors draw and show some profiles, e.g. for the data shown in Fig. 2?\n- what was the effective repetition rate in the experiment and how many shots does it take to have a full image of the dynamics? Besides, I would not say that 1 Hz is a high repetition rate for such a system (it would have been for a real ICF experiment). In ref. 37 for instance, the repetition rate was in practice 0.3 Hz - i.e., the same order of magnitude - for the case of a solid target. Thus I suggest to remove or soften the mentions about the high-repetition rate operation.\n- There is no laser filter in the setup depicted in Fig. 1. Where was it located?\n- For the phase contrast imaging application in the field of HED and ICF experiments, it would be interesting to mention and discuss a different platform composed of a nanosecond X-ray backlighter coupled with a streaked camera. Such setup produces nice results at NIF - see e.g. Physical Review Letters 129.21 (2022): 215003, for the tracking of fuel-ablator interface trajectory as well as the shock evolution inside the ablator in ICF implosions. Given the relatively long (nanosecond) dynamics and time scale of such physical processes, is it really relevant to use a femtosecond source, while an X-ray streak camera can provide <100 ps time resolution - and a gated full-frame detector around ~1 ns or less?\n- The electron beam has a very broad spectrum. Can the Authors comment on the effect of this energy spread when probing electromagnetic fields in the sample? There should be important chromatic effects in the focusing/defocusing trajectories of the e-beam through the sample. Can't it lead to misinterpretation of the results?\n- aren't there some drawbacks to this dual probe? In particular, since the e-beam crosses the sample and probably its mechanical mount, isn't it an additional source of noise on the X-ray detector?\n- line 131 and section 4.5: I would write \"gas target\" instead of \"plasma target\"\n- line 239: what does justify a polytropic index value of 5/3?\n- in Methods, eq. 5: could the Authors provide a reference?\n- Section 4.8: what is the physical justification of using a radial and an elliptical frequency mask?\nMoreover, I noticed few typos:\n- line 314/315: \"in the of order\"\n- line 640: \"by by\"\n- in ref 9 : DOI is https://doi.org/10.1063/1.3485109\nIn conclusion, I would suggest some revisions and answers to the points listed above before publication.", "reviewer_3": "Multi-messenger dynamic imaging of laser-driven shocks in water using a plasma wakefield accelerator\nSummary and General Comments:\nThis paper presents data combining ultrafast betatron X-ray imaging and relativistic electron beam radiography, both generated by a laser wakefield accelerator to investigate the dynamics of laser-driven shocks in a water jet target. The authors demonstrate that this “multi-messenger” approach provides complementary insights into hydrodynamic shock formation and associated electromagnetic field evolution, which are not accessible with a single probe diagnostic. The work addresses important challenges in high-energy-density physics, especially in the context of high rep-rate plasma/target diagnosis.\nBy utilizing both ultrafast X-ray imaging along with electron beam radiography, the study captures detailed shock dynamics and intricates field structures. Precise calibration of both diagnostics ensures accurate quantitative analysis. Additionally, the use of FLASH simulations reinforces the experimental findings by replicating the shock morphology and clarifying the impact of the surrounding vapor layer.\nOverall, this manuscript presents sound results, is well-written and merits publication in Nature Communications.\nSuggestions for Improvement:\nComment: The authors state the shock breakout occurs at ~ 2 ns, which aligns with the measured shock velocity and reverse shock appearance time. However, the arrow in Figure 2c indicates breakout at around 4.5 ns. Please correct this discrepancy. Additionally, Figure 2a has a significant gap in the image sequence between 0.82 ns and 4.5 ns. Are images available at approximately 2 ns? If so, they should be included. Similarly, for the image sequence in Figure 5.\nComment: Figure 5 presents measurements and simulations of shock velocity up to 1 ns. I recommend including data points within the 1-2 ns interval (the breakout time) if exist, to better illustrate the linear increase up to the breakout time.\nComment: The author states: “FLASH simulations accurately model the fluid dynamics but do not include electric (E) and magnetic (B) fields nor treat hot electron populations.” p.12, l.581. Please expand the discussion about the impact of the hot electron populations in the present study (mainly for the broader audience).\nComment: Provide further information on the sensitivity and limitations of the field recovery method, especially in regions with caustics or large deflection angles, to better show its range of applicability. The author states: “However, in the regime where the deflection angles αₓ and αᵧ are large, the fields become too strong, leading to particle crossing along the trajectories...– limiting the applicability of the algorithm for inversion of radiographs.” p.15, l.740\nCould the authors expand on how these limitations affect the recovered field magnitudes and the overall applicability of the method?\nComment: Please elaborate on the mechanism by which the low-density vapor layer influences shock symmetry.\nAdditionally, provide a brief summary of the parameters used in the Fourier and Meijering filtering techniques for image processing. The author states: \"Including the low-density layer in the simulations enhances thermal transport around the water column, resulting in more uniform heating of the jet's exterior and producing a cylindrical shock and symmetric compression morphology closely matching the experimental observations\" (p. 6, line 300).\nSpecifically, explain in greater detail how this low-density vapor layer contributes to uniform heating and the resulting shock symmetry. Furthermore, outline the parameters employed in the Fourier and Meijering filtering of the images.\nComment: This isn't a comment, but rather a point of discussion. The Atwood number is 3/5 suggesting that the development of Richtmyer-Meshkov and later Rayleigh-Taylor instabilities are possible. Given typical growth rates for the experiment described, the instability should be detectable in Figure 2's data within a few nanoseconds, with amplitudes reaching several microns (a few pixels in the images). If the resolution is sub-micron, these perturbations should be visible, especially at later times. Have data been collected that demonstrate density perturbations consistent with the Richtmyer-Meshkov instability? A tooth-like pattern is discernible in the data. Does the data at ~ 8 ns show an expanded version of this pattern? Please comment in the text.\nComment:\nAgain, this isn't a comment, but rather a point of discussion. While the experiment's diagnostic value is clear to experts, it is also clear that such an experimental layout is difficult to be implemented in an existing laser-fusion facility such as for instance NIF. However, this study offers significant value for future facility designs.\nComment:\nThe current literature on betatron generation via lasers within this article is limited. Additional references utilizing laser systems of comparable power should be included. Specifically, I recommend the inclusion of at least the following two references, both of which focus on optimizing betatron radiation generation via Laser Wakefield Acceleration (LWFA):\nGrigoriadis, A., Andrianaki, G., Tatarakis, M., Benis, E. P., & Papadogiannis, N. A. (2021). Betatron-type laser-plasma x-ray sources generated in multi-electron gas targets. Applied Physics Letters, 118(13).\"\nYe, H., Gu, Y., Zhang, X., Wang, S., Tan, F., Zhang, J., ... & Zhou, W. (2022). Fast optimization for betatron radiation from laser wakefield acceleration based on Bayesian optimization. Results in Physics, 43, 106116.\nMinor Note on Language:\nThere is a typo “chromatogrpahy” should be “chromatography.” p.12, l.586", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to Reviewers\nManuscript Title: Multi-messenger dynamic imaging of laser-driven shocks in water using a\nplasma wakefield accerlerator\nCorresponding Author: Mario D. Balcazar, balcazar@umich.edu\nManuscript Number: NCOMMS-24-86275\n1. Reviewer #1 (Remarks to the Author):\nThe paper reports on laser-driven shocks in water, observed through secondary sources generated\nby a laser plasma accelerator.\nFirstly, I find this work highly impressive and original. The quality of the data is very good, and\nthe choice of diagnostics is highly relevant. It is particularly exciting to see shock propagation\ncharacterized at such a high repetition rate. This achievement is the result of the careful selection\nof the laser system, target, and X-ray betatron and electron beams, which together made this\nwork possible.\nResponse:\nWe thank the reviewer for the supportive comments.\nThe article presents interesting experimental data and numerical results of high quality. While the\nsuggested explanations appear plausible, additional analysis and numerical simulations would\nfurther strengthen the conclusions. The following comments need to be addressed for the paper\nto be accepted.\n1.1 Question 1: The data related to shock propagation as shown in Figure 2 from the X-ray\nimages indicated a prompt “compression”: What is the origin of the signals that appear at all\ntimes (except at 0 ns) ahead to the right (and sometimes to the left) of the shock?\nResponse:\nWe appreciate the reviewer question pointing out these features in Figure 2 that might have not\nbeen sufficiently clarified. The X-ray signals that appear ahead of the shock, both on the left and\nright, precisely are attributed to the ‘shock’ compression from the back side of the target. As it is\nlater detailed in Figure 4. The electron heat transport following the laser energy deposition\nresults in a rather symmetrical heating of the target, and even though the laser-facing-side results\nin a stronger shock generation, the back side of the target also experiences high temperatures and\npressures resulting in a secondary shock production which, as you point out, can be discerned\nearly in time and persists through the whole shock propagation and target compression. We have\nupdated the text to improve clarity regarding these features.\nOriginal:\n● Results section: p. 4 line 198-203:\n○ “ The laser-driven shock displays very similar features in both simulation and\nexperiment, including a bow-like shape that develops vertically and propagates\nforward horizontally.’’\nModified to:\n● Results section: p. 4 lines 201-204:\n○ “The features observed in Fig. 2 include bright phase-contrast enhancement at the\nedges of the target, as well as a strong, dark, bow-shaped shock structure that\ngrows in y-direction and propagates forward in x-direction. Smaller precursor\nsignals ahead of the main shock are also appreciable and are explored further in\nthe results.’’\n1.2 Question 2: Since the relevant data exist, it would be beneficial to include the corresponding\nnumerical data in Figure 2c. This addition would help track the comparison from t = 0 up to a\nfew nanoseconds. As it currently stands, there is a gap in the information between 0.83 ns and\nabout 8 ns (with only one time step at 4.5 ns).\nResponse:\nWe thank the reviewer for this helpful suggestion. In response, we have now included the\ncorresponding simulation data (Fig. 2b) alongside the experimental image sequence (Fig. 2c),\nenabling a direct side-by-side comparison of the shock evolution over time. To accommodate this\naddition, the previous Fig. 2b has been moved to become the new Figure 9a. This update might\noffer a more complete and continuous view of the dynamics captured in both simulation and\nexperiment. We also reassessed the use of 10-shot averaged images for lineouts in Fig. 2c and\nfound that single-shot lineouts could better capture the spatial features in the composite image.\nAccordingly, we have updated Figure 2c as shown below.\nOriginal:\n● Fig. 2c:\n● Fig. 2 Caption:\n○ “Dynamic betatron X-ray imaging of laser-driven shock in water and comparison\nwith 3D FLASH simulations. (a) Contrast between synthetic phase-contrast X-ray\nimages and experimental betatron X-ray images. (b) Illustration of mapping\nbetween FLASH projected density distributions to synthetic phase-contrast X-ray\nimages with Fresnel-Kirchoff algorithm. (c) Average lineouts taken at the center\nof the betatron images and stacked together horizontally to obtain a composite\npicture of the full delay (8 ns) of the interaction.”\nModified to:\n● Fig. 2b and Fig. 2c:\n● Fig. 2 Caption:\n○ “Comparison between betatron X-ray imaging and 3D FLASH hydrodynamic\nsimulations of laser-driven shocks in water. a Contrast between experimental\nbetatron X-ray images and synthetic phase-contrast X-ray image (see Sec 4.8.2) at\ndifferent time delays. b and c Lineouts averaged over 80 pixel rows taken at the\ncenter of simulation and experimental images, and stacked together horizontally\nto obtain a composite picture of the full delay of the interaction. Cyan dots\nindicate tracking of shock position.”\n1.3 Question 3: Why is the ablated plasma not visible at all? One would expect to observe a\nplasma plume, at least during the duration of the heating laser pulse.\nResponse:\nWe thank the reviewer for this insightful question. We assume this refers to the apparent absence\nof the plasma plume in the X-ray images. As the reviewer suggests, one might expect to observe\nablated plasma expanding into vacuum during the duration of the nanosecond laser pulse.\nHowever, several factors limit its visibility in the betatron radiographs.\nTo begin with, we estimate the electron density (~1023 cm-3 ) and temperature (~ 200 eV) of the\nablated plasma plume using our FLASH simulations. These suggest that the corresponding\npressure is about (~104 Torr). Utilizing the CXRO toolkit\n(https://henke.lbl.gov/optical_constants/) for X-ray transmission through a gas, it suggests that\nfor our X-ray critical energy (~4 keV) the transmission exceeds 98%.\nFurthermore, and perhaps more importantly, given that our geometrical setup is set for\npropagation-based X-ray phase-contrast imaging, the density gradients play a major role in the\nproduction of the image. In this sense, the density gradients within the expanding plasma are\nrelatively smooth and gradual, several orders of magnitude smaller than those present at the\nedges of the target or for the strong shock-generated gradients. Such small gradients lead to only\nminimal phase-shift or deflection angles, well below the angular resolution threshold of our\nimaging system. As a result, the X-ray probe is largely insensitive to this low-density plasma\nstructure.\nThis very limitation underscores the motivation for our multi-probe approach: although the X-ray\nimages do not reveal the ablated plasma, the relativistic electron beam is highly sensitive to\nintegrated electromagnetic field gradients associated with the plume. This is evidenced in Figure\n5, where field-driven deflections in the electron beam directly visualize the ablated plasma\nexpansion and structure that remain invisible to the X-ray probe.\n1.4 Question 4: The linear scale in Figure 4b should be replaced with a logarithmic scale (as\ndone in Figure 9 for the temperature). This change would facilitate a better understanding of\nplasma expansion and ablation.\nResponse:\nWe thank the reviewer for the suggestion. While a logarithmic scale for Fig. 9 (now Fig. 10) is\nindeed useful for visualizing the electron temperatures across multiple orders of magnitude, we\nbelieve that in the case of the strong shock surrounding the water jet in Fig. 4b, it may actually\nhinder in depicting the main argument of the figure. Given that the strong shock leads to a mass\ndensity and pressure increase of less than a factor of 10 (compared to nominal density and\npressure of water) a linear scale helps in depicting the evolution of the shock more clearly, while\nit gets lost when using a logarithmic scale for this particular figure. In the case of the temperature\nplot, the variation in temperature around the target spans several orders of magnitude, making a\nlogarithmic scale is more appropriate when displaying this simulation features. The comparison\nof Figure 4b with and without logarithmic scale is shown below.\n● Linear Scale:\n● Logarithmic Scale:\n1.5 Question 5: The electron distribution appears quite surprising. What is the origin of the two\nmain peaks at around 45 MeV and 170 MeV? A continuous injection and broad spectrum would\ntypically be expected. This anomaly should be clarified and explained through numerical\nsimulations and/or by considering instrumental responses.\nResponse:\nWe appreciate the referee’s insightful observation. This prompted us to identify an error in the\nspectrometer recording during the original scan. The magnetic spectrometer instrument used in\nthe experiment is composed of four different cameras, requiring the full spectrum to be stitched\ntogether using individual trigger signals. Figure (a) below illustrates the spectrometer geometry,\nshowing the trajectories of electrons with various energies bending toward the Lanex screen.\nCameras 1 and 2 view the forward screen, while Cameras 3 and 4 capture the bottom screen.\nFigure (b) shows a sample reconstructed single-shot electron spectrum by stitching images across\nfour cameras, with the following (calibration dependent) energy coverages:\n● Camera 1: 175–500 MeV (original) / 150–500 MeV (revised)\n● Camera 2: 66–175 MeV (original) / 60–150 MeV (revised)\n● Camera 3: 43–58 MeV (original) / 35–60 MeV (revised)\n● Camera 4: 0–43 MeV (original) / 0–35 MeV (revised)\nRegarding the observed features in the original spectrum: the dips or gaps at approximately 43\nMeV, 58–66 MeV (gap), and 175 MeV correspond to stitching boundaries between adjacent\ncamera ranges, confirming that these are stitching artifacts rather than physical spectral features.\nThe true main peak of the beam is near 170 MeV, captured by both Camera 1 and Camera 2. The\npeak around 45 MeV was not accurately captured in the original scan due to a camera stitching\nand triggering mismatch, which led to the low-energy tail being cut off.\nTo address this, we have identified an alternative dataset acquired under the same experimental\nconditions, in which the low-energy tail is properly recorded. We have replaced Fig. 7 in the\nrevised manuscript with the corrected spectrum shown below.\nDespite the presence of stitching discontinuities (also visualized in the revised Fig. 7), the\nrevised spectrum clearly exhibits a main peak near 150 MeV, along with a prominent low-energy\ntail. Importantly, low-energy electrons, particularly those below 20 MeV, have much larger\ndivergence and often fall outside the 20 mrad acceptance angle of the magnetic spectrometer,\nresulting in their underrepresentation in the measured spectrum.\nAccording to previous simulations (Refs. 1, 2, 3 in the response), the low-energy tail is most\nlikely composed primarily of self-injected electrons generated near the exit of the gas jet, which\nfeatures a 1 mm downramp. In contrast, the main energy peak arises from ionization injection\nalong the full 6 mm plasma length.\nReferences:\n● [1] M. Hansson et al., PRST-AB 18, 071303 (2015);\nhttps://journals.aps.org/prab/pdf/10.1103/PhysRevSTAB.18.071303\n● [2] D. P. Grote et al., Phys. Plasmas 28, 123104 (2021);\nhttps://doi.org/10.1063/5.0073622\n● [3] S. Kuschel et al., Phys. Rev. Lett. 121, 154801 (2018)\nhttps://journals.aps.org/prl/pdf/10.1103/PhysRevLett.121.154801\nOriginal:\nModified to:\nReferences:\n● [57] Tsai, H.-E., Swanson, K.K., Barber, S.K., Lehe, R., Mao, H.-S., Mittelberger,\nD.E., Steinke, S., Nakamura, K., Tilborg, J., Schroeder, C., et al.: Control of\nquasi-monoenergetic electron beams from laser-plasma accelerators with\nadjustable shock density profile. Physics of Plasmas 25(4) (2018)\nhttps://doi.org/10.1063/1.5023694\n● [58] Nakamura, K., Gonsalves, A., Lin, C., Smith, A., Rodgers, D., Donahue, R.,\nByrne, W., Leemans, W.: Electron beam charge diagnostics for laser plasma\naccelerators. Physical Review Special Topics—Accelerators and Beams 14(6),\n062801 (2011) https://doi.org/10.1103/PhysRevSTAB.14.062801\n1.6 Question 6: The unit for the electron energy distribution could be changed to pC/MeV.\nResponse:\nWe thank the reviewer for catching this typo. The correct unit for the y-axis of the electron\nenergy spectrum is indeed pC/MeV, and this has now been corrected in the revised Fig. 7. Since\nthe x-axis is expressed in MeV (rather than MeV/c), the units are now consistent, such that\nintegrating the distribution yields the total beam charge in pC. We appreciate the reviewer’s\ncareful attention to detail.\n1.7 Question 7: The analogy with low-density foam is very interesting. However, for it to be\nfully relevant, a comparison between the density of the foam used in fusion experiments and the\nvapor density employed in the FLASH simulations would be necessary. It appears that the vapor\ndensity used in FLASH simulations may be of several orders of magnitude lower than that of the\nfoam.\nResponse:\nWe thank the reviewer for this useful comment. A new reference has been added (substituted) in\nthe text (Bhandarkar et al., 2017) that describes the fabrication of ICF targets with foam lined\nHohlraums at Lawrence Livermore National Lab (LLNL). Following this reference the foam\nliner inside the Hohlraums used in ICF experiments needs to have a density in the order of 20\nmg/cc or lower for optimal performance. Even though ideal performance of the foams seems to\nbe achieved at rather low densities, they wouldn’t want to keep the density of the foam liner\nlower than necessary in order to increase the handling yield. Furthermore, previous Ref. [52] of\nthe manuscript experiments are described with Au-doped CH and Ta2O5 foams with densities\nbetween 10 - 40 mg/cc.\nFor the 3D FLASH simulations presented in our work the ‘foam-like’ vapor layer surrounding\nthe dense water target with radius r , defined in the region where r> r , has a density profile with\n0 0\nshape\nρ = 𝑛 (𝑟 /𝑟) for r > r\n𝑐 0 0 0\nWith an initial vapor density n at the edge of the target. For the results presented in our work, we\n0\nuse n = 10 mg/cc, which closely matches the description in Bhandarkar et al. and Ref.[52] for\n0\nfoam liners used in ICF targets. Moreover, according to these references, the thickness of\n(density dependent) ICF foam liners is typically on the order of hundreds of micrometers. At a\ndensity of n = 10 mg/cc, the corresponding foam thickness would be approximately 500 um,\n0\nwhich is also consistent with our experimental observations in Figure 5, and around the same\norder of magnitude of the relevant area influencing the compression of the target in the FLASH\nsimulations. Choosing the specific value of n = 10 mg/cc for the simulations was decided to\n0\nmatch the behavior observed in the experimental results in Figure 4. At significant lower\ndensities, compression is predominantly one-sided, and the vapor assisted ‘foam-like’ cylindrical\ncompression effect is no longer observed.\nFinally, Even though previous reference [53] mentions a higher density foam in the order of 800\nmg/cc, a reason for this might be due to the fact that the study is mostly focused on\ndemonstrating the 3D Ptychotomography technique at LCLS, and using nanofoams samples with\nsimilar morphological characteristics ICF foams. Moreover, the large increase in density used\nand lower thickness is potentially be due to manufacturing constraints for the use of copper in\nthis particular experiment, or to facilitate easier handling and installation of the target to be used\nwith the XFEL.\nTo more clearly highlight the relevance of the vapor layer used in our FLASH simulations to the\nfoam densities typical of ICF experiments, we have revised the manuscript as follows.\nOriginal:\n● Discussion section p. 10 line 497 - 506:\n○ “Through experiments conducted at the National Ignition Facility, these liners\nhave demonstrated effectiveness in mitigating wall motion that otherwise\ninterferes with a fully spherically-symmetric compression of the D-T fusion\ntarget. In an analogous sense, the low density water vapor layer in this experiment\nserves to assist the single laser ablator to compress the target in a symmetrical\nfashion.”\n● References section:\n○ [53] Carr, A., Sechrest, Y., Mertes, K., Patterson, B.M., Wohlberg, B., Hancock,\nL., Sirica, N., Sandberg, R., Sweeney, C., Hunter, J., et al.: Morphology of copper\nnanofoams for radiation hydrodynamics and fusion applications investigated by\n3d ptychotomography. Nano Letters 24(32), 9916–9922 (2024) https://\ndoi.org/10.1021/acs.nanolett.4c02289\nModified to:\n● Discussion section p. 10 line 470 – 479: :\n○ “Previous experiments have demonstrated effectiveness in mitigating wall motion\nthat interferes with a fully spherically-symmetric compression of the D-T fusion\ntarget through liners with densities in the order of ρ = 0.01–0.04 g cm-3 . In an\nanalogous sense, the low density water vapor layer with ρ = 0.01 g cm-3 in this\n0\nexperiment serves to assist the single laser ablator to compress the target with a\ncylindrically-symmetric morphology.”\n● References section:\n○ [53] Bhandarkar, S., Baumann, T., Alfonso, N., Thomas, C., Baker, K., Moore,\nA., Larson, C., Bennett, D., Sain, J., Nikroo, A.: Fabrication of low-density foam\nliners in hohlraums for NIF targets. Fusion Science and Technology 73(2),\n194–209 (2018) https://doi.org/10.1080/15361055.2017.1406248\n1.8 Question 8: The explanation of the role of different plasma species could be strengthened by\nincluding FLASH simulation results showing the distributions of electrons, protons, and oxygen\nspecies.\nResponse:\nWe thank the reviewer for this suggestion. In FLASH, material species such as water are\nmodeled as effective, indivisible atoms characterized by their average atomic mass and charge\nsee Ref. . Rather than treating water as distinct hydrogen and oxygen species, it is represented as\na single composite fluid species with an initial atomic mass number A = 6 and charge number Z\n= 3.3, corresponding to the weighted averages of its molecular parts.\nA typical FLASH simulation utilizes the ‘Simulation initBlock’ section to initialize mass scalar\nvariables ‘sumy’ = ∑𝑌 and ‘ye’ = Y . From which the local average atomic number\n𝑖 e\n𝐴 = 1/∑𝑌 and average ionization state 𝑍 = 𝑌 × 𝐴 are derived. These quantities evolve\n𝑖 𝑒\ndynamically and vary spatially, enabling inference of free electron distributions, though not\nseparated by atomic species.\nTo help visualize ionization evolution, below we included a figure showing the spatial\ndistribution of 𝑍 at multiple time steps. This provides insight into the changing degree of\nionization across the shock and ablation region, however is unable to discern between individual\nprotons and oxygen ions.\nWhile some of these ion species independent effects could potentially be captured by PIC\nsimulations, the nanosecond timescales makes this approach rather unsuitable as ‘grid heating’ is\nan important limitation in PIC simulations at large (nanosecond) timescales. Perhaps some\nhybrid simulation models could potentially capture this, yet the authors don’t have access to\nthese.\nImportantly, the FLASH modeling limitation further highlights the utility of the electron beam\nprobe, which is sensitive to electromagnetic field structures from charge-separation and\ndifferentiation in ion species, not resolved by FLASH and neither captured by X-ray imaging.\nThis reinforces the complementary nature of the approach introduced in our study.\nReferences:\n[47] Fryxell, B., Olson, K., Ricker, P., Timmes, F.X., Zingale, M., Lamb, D.Q., MacNeice, P.,\nRosner, R., Truran, J.W., Tufo, H.: Flash: An adaptive mesh hydrodynamics code for modeling\nastrophysical thermonuclear flashes. The Astrophysical Journal Supplement Series\n131(1), 273 (2000) https://doi.org/10.1086/317361\n1.9 Question 9: Figure 9 shows sharp gradients and unusual features, such as the rectangular\nshape in the first image extending from in microns -250 to 0 along the x-axis and from -100 to\n100 along the y-axis. Other sharp gradients visible in the images should also be discussed.\nResponse:\nWe thank the reviewer for identifying this issue. The sharp gradients observed in the original Fig.\n9 were the result of an error in the laser setup parameters used in the FLASH simulations.\nSpecifically, the Gaussian laser profile was inadvertently truncated too early due to an incorrect\nsetting in the input function. This led to an artificially abrupt edge in the simulated profiles and\nintroduced rectangular artifacts in the temperature profiles.\nWe have addressed this by extending the truncation range, which smooths out the artificial\nboundaries and eliminates the sharp gradients as shown below,\nThe updated version of Fig. 9 (now Fig. 10) reflects this correction and has been included in the\nrevised manuscript. We are grateful to the reviewer for bringing this to our attention.\nOriginal:\n● Figure 9:\nModified to:\n● Figure 10:\n1.10 Question 10: Why justify the choice of the flux limited diffusion coefficient to 0.06, and\nhow the results change with other values?\nResponse:\nWe thank the reviewer for this insightful question. For a specific target, the value of the flux\nlimiter 𝑓 is typically not derived from first principles but rather estimated empirically based on\nprior experimental results. As discussed in Chen et al. (2023) (now Ref. [65] in the revised\nmanuscript), for planar targets 𝑓 is usually taken to be between 0.03 − 0.1, while for spherical\ntargets, values in the range 0.06 − 0.5 are more typical. The appropriateness of a given\nflux-limiter value is closely tied to the specific laser intensity and wavelength being used.\nIn our case, the shock-driving laser is a Ti:sapphire system (λ = 800 nm, energy = 0.5–1 J,\npulse duration = 200–300 ps, spot diameter = 40 μm), corresponding to intensities on the order of\n14\n𝐼 > 10 W/cm2, where non-local transport effects start to become significant. Chen et al.\nbenchmark the Schurtz–Nicolai–Busquet (SNB) non-local heat transport model and compare it\nwith flux-limited diffusion. They find that for lasers with similar intensities and wavelengths,\nbetween 1ω (1000 nm) and 2ω (500 nm), a flux-limiter in the range of 0.14 − 0.20 provides a\nbetter approximation to non-local behavior than lower values. Based on this, we conclude that\nour initial choice of 𝑓 = 0.06 is likely suboptimal, and a more appropriate value for our\nconditions is 𝑓 = 0.17.\nTo assess the sensitivity of our results to the choice of flux-limiter, we ran a series of FLASH\nsimulations using 𝑓 = 0.01, 0.06, 0.10, 0.17, 0. 30, 0.50, 0.80, as shown below.\nAcross most of these cases with 𝑓 = 0.06 – 0.80 we observed only modest variations in the\nshock propagation velocity. The robustness of the shock dynamics suggest that energy deposition\nand the resulting pressure drive remain the dominant factors, and that variations in thermal\ntransport do not substantially influence the key hydrodynamic behavior relevant to our findings.\nNevertheless, based on the analysis in Chen et al. and taking into account our laser parameters,\nand simulation results, we believe that a flux-limiter value of 𝑓 = 0.17 offers a more physically\nappropriate approximation for our experimental conditions. Accordingly, we have updated the\nsimulations, text and relevant figures to a more appropriate dataset. While this change improves\nthe theoretical consistency of our modeling, the difference in shock evolution compared to f =\n0.06 remains within the same order of magnitude and does not affect the core results or\ninterpretations presented in the manuscript.\nOriginal:\n● Methods section p. 12 Line 601–602:\n○ …and the multi-group flux-limited diffusion coefficient was chosen to be\n𝑓 = 0.06.\nModified to:\n● Methods section p. 13 Line 658–659:\n○ … and the multi-group flux-limited diffusion coefficient was chosen to be\n𝑓 = 0.17, considering our laser parameters and Ref.~\\cite{chen2023effect}.\n● Fig. 3 updated\nReferences (addition):\n● [65] Chen, Z.H., Yang, X.H., Zhang, G.B., Ma, Y.Y., Xu, H., Luan, S.X., Zhang, J.:\nEffect of non-local transport of hot electrons on the laser-target ablation. Physics of\nPlasmas 30(6) (2023) https://doi.org/10.1063/5.0146342\n2. Reviewer #2 (Remarks to the Author):\nThe manuscript presents results of a pump-probe experiment where relativistic electrons and\nbetatron X-rays produced with a laser-plasma accelerator are used to probe the dynamics of a\nwater sample heated by a secondary laser pulse.\nThe X-rays provide a radiograph of the shock front. In parallel, the relativistic electron beam is\nused to qualitatively track electromagnetic fields in the plasma formed by the laser-water\ninteraction.\nThe main achievements or points of novelty are:\n● This is a real pump-probe application, while many recent articles using betatron as a\nprobe have been limited to \"cold\" samples, without any dynamical study.\n● The application to shock propagation imaging. Up to now, the use of betatron X-rays for\nthis application was restricted mainly to the article of Wood et al. Scientific Reports\n2018. Here, there is moreover a deeper analysis of plasma conditions and the results are\nsupported by hydrodynamic simulations.\n● The use of liquid water as a sample, which has an interest for some range of high-energy\ndensity experiments. It is also technically appealing since it allows for continuous\nrenewal of the target\n● The use, in combination to X-ray radiography, of the electron beam to probe the plasma.\n● The analysis, though it includes important steps of image processing and data filtering,\nseems consistent. I am however not a specialist of laser-generated shocks in matter, and I\nhope that other reviewers are more experts of this physics.\nIn my opinion, the work will be of significance to the field and deserves publication in a high\nimpact journal.\nResponse:\nWe thank the reviewer for the supportive comments.\nHowever, I have some concerns and questions:\n2.1 Question 1: While reading the manuscript, I had sometimes the feeling that the Authors try\nto oversell the results - whereas I do not think it is useful. The repetition and highlights all along\nthe text of the \"multi-messenger approach\", and some specific sentences such as \"The\nsimultaneous inclusion of ultrafast X-rays and an electric/magnetic field-sensitive probe in\nLWFA introduce a novel multi-messenger imaging technique\" in the introduction, imply a\ntechnological breakthrough and/or a novel scientific approach, which is not really the case. The\nfact that the e-beam probe can complement the X-ray imaging is indeed interesting, but it is not a\nparadigmatic change.\nResponse:\nWe value the reviewer’s constructive feedback and agree that avoiding redundancy in\nterminology throughout the text is important. In response, we have revised the manuscript to\nsoften the emphasis on the “multi-messenger” approach. Specifically:\nOriginal:\n● Abstract section p. 2 line 63 – 65:\n○ “This multi-messenger approach highlights the need for hybrid physics models\nthat integrate the detailed insights from both probes, spanning kinetic to\nhydrodynamic effects, to accurately predict fusion-relevant plasma dynamics from\nfemtosecond to nanosecond timescales.”\n● Introduction section p. 3 line 114 – 119:\n○ “The simultaneous inclusion of ultrafast X-rays and an electric/magnetic\nfield-sensitive probe in LWFA introduce a novel multi-messenger imaging\ntechnique, uncovering experimental details that would remain hidden from using\na single probe alone.”\n● Results section p. 3 line 141-142:\n○ “…providing a unique tool for multi-messenger imaging...”\n● Discussion section p. 10 line 485 – 491:\n○ “Moreover, the multi-messenger technique may also motivate the construction of\nultrashort pulse laser systems capable of LWFA diagnostics directed to large scale\nfusion facilities such as the National Ignition Facility, Laser MegaJoule or LLE\nOMEGA.”\nModified to:\n● Abstract section p. 2 line 62–64:\n○ “We show that combining both probes provides complementary insights spanning\nkinetic to hydrodynamic regimes, highlighting the need for hybrid physics models\nto accurately predict fusion-relevant plasma behavior.”\n● Introduction section p. 3 line 109 – 113:\n○ “In this work we show that utilizing both, high-resolution X-ray photons and\ncoordinated electromagnetic field-sensitive electrons, one can reveal experimental\ndetails that would remain hidden from using a single particle type alone”\n● Results section p. 3 line 138 – 139:\n○ “…providing a unique tool for imaging...”\n● Discussion section p. 11 line 522 – 527:\n○ “These insights may also motivate the design of future smaller-scale HED science\nexperiments, which could one day be adapted to large-scale fusion facilities such\nas the National Ignition Facility, Laser MegaJoule, or OMEGA.”\n2.2 Question 2: Moreover, the experimental layout relies on placing the sample before the dipole\nmagnet instead of placing it after, which is not very complicated (and is usually done to avoid\nperturbation of the sample itself or noise in the X-ray detectors). I also guess that such layout\nwas driven by geometrical considerations for phase-contrast imaging criteria and magnification\nconsiderations. I suggest that the Authors \"de-emphasize\" the message on the multi-messenger\napproach.\nResponse:\nThank you to the reviewer for pointing this out. Indeed, the experimental geometry —\nspecifically placing the sample before the dipole magnet — was primarily determined by two\nkey requirements: (1) satisfying the conditions for propagation-based phase-contrast X-ray\nimaging, which imposes constraints on the source-to-object and object-to-detector distances to\nensure sufficient magnification is achieved to resolve sub-micron features in the shock structure.\n(2) The minimum distance from the source is acquired to develop X-ray spatial coherence, as\ndescribed in the Methods Sec. 4.1. These considerations were the major factors that dictated the\nplacement of the sample within the beamline. We agree that this geometric configuration, while\nintentionally chosen to enable the dual-probe implementation, does not in itself constitute an\nexperimental innovation. That said, we believe that the layout simplicity might be an advantage\nfor reproducibility and accessibility of the experiment.\nWe have further toned down the emphasis on the “multi-messenger” approach throughout the\ntext, as described in the response to Question 2.1.\n2.3 Question 3: In the same vein, I suggest to remove the first sentence of the abstract, because\nthis is not (and far from) an ICF experiment, and it is probably not the main motivation at this\npoint.\nResponse:\nWe thank the reviewer for their suggestion and acknowledge that the opening sentence of the\nabstract may overstate the connection to ICF ignition experiments. Our motivation stems from\nbroader challenges in diagnosing high-energy-density systems relevant to ICF, rather than from\nattempting to replicate an ignition-scale experiment. In response, we have revised the first\nsentence of the abstract to more accurately reflect this broader context:\nOriginal:\n● Abstract section p. 2 line 53 – 54:\n○ “Understanding dense fuel hydrodynamics is critical for predicting burning\nplasma behavior in laser-driven inertial confinement fusion.”\nModified to:\n● Abstract section p. 2 line 53 – 54:\n○ “Understanding dense matter hydrodynamics is critical for predicting plasma\nbehavior in environments relevant to laser-driven inertial confinement fusion.”\n2.4 Question 4: Nevertheless, I completely agree with the last two sentences of the conclusion,\nabout the perspectives of this work in the field of ICF, HED science and for the development of\nsuch sources and diagnostics at large-scale fusion facilities.\nResponse:\nWe appreciate the reviewer’s comment recognizing the broader perspective outlined in our\nconclusion. A central goal of this work is to demonstrate how compact, dual-probe laser\nwakefield accelerator systems can advance diagnostic capabilities in high-energy-density science\nand potentially inform future strategies at large-scale fusion facilities. We are pleased that this\noutlook was well received.\n2.5 Question 5: There is no discussion about the repeatability and data statistics. Were all the\nmeasurements taken in a single shot (for a given delay and a given probe source)? What about\nthe repeatability of the measurements ? Since the sample is fluid, isn't it difficult to reproduce\nshot-to-shot the same conditions? (shock front position, EM-fields, etc.). It would be interesting\nto see a video or a bunch of pictures of the shock images, taken at the same laser and delay\nconditions. The stability of the observed phenomenon is crucial here, especially as the two\nprobes do not record the same shots.\nResponse:\nWe thank the reviewer for raising this important point. One of the most compelling aspects of\nthis experiment is its high degree of repeatability, both within a single run and across different\ndays or even months. Under stable laser and target conditions, we are able to acquire reliable\nmeasurements using single-shot exposures at each delay position. While the water target is a\nfluid system and therefore subject to shot-to-shot variation in principle, we found that the shock\ndynamics remained highly reproducible, owing to the stability of both the water column and the\nlaser heater beam.\nTo illustrate this, we have now included sets of X-ray images taken under the same laser and\ndelay conditions for three representative time steps, showing the consistency of the shock front\nmorphology and evolution.\nWhile it is true that small variations (as undulations) can be appreciated along the water column\nfrom its liquid nature, it seems like the shock dynamics are sufficiently robust to be repeatable.\nRegarding the electromagnetic field measurements obtained with the electron probe, we note that\nthese are also similarly repeatable. While we would not expect exact replication in the fields due\nto the complex interplay in laser energy absorption, electron heat transport and shock generation,\nthe overall structure and topology of the recovered fields are consistent across repeated shots.\nThe largest source of variation lies in the electron beam probe itself, which is susceptible to\nfluctuations in the gas target conditions during the laser wakefield acceleration process. Even so,\nthe observed field features and underlying dynamics are robust, as demonstrated in repeated\nshots presented below for the same delays.\nThese results underscore the strength of our approach in which despite recording independent\nshots for the X-ray and electron diagnostics, the data reveal a coherent and stable picture of the\nevolving plasma dynamics.\nWe have added a note in the discussion regarding this repeatability, and we are happy to add a\nFigure to the Extended Data section for this matter if the editors feel that is warranted.\nModified to (addition):\n● Discussion section p. 10 Line 505 – 511:\n○ “All measurements were repeated across multiple shots (typically 5–10 per delay\ntime) under the same conditions. Despite the fluid nature of the target, key\nfeatures such as shock evolution, compression morphology, and field topology\nwere reasonably reproduced across the dataset.”\n2.6 Question 6: Was there a background substration on the X-ray imaging measurements? If yes,\nwas it taken with or without water flow, with or without accumulation on several shots?\nResponse:\nYes, we thank the reviewer for pointing this out. Background subtraction was performed for all\nX-ray imaging measurements. To obtain a representative background, we recorded images with\nno water flow and averaged 10 shots to improve the signal-to-noise ratio. This averaged\nbackground was subtracted from each signal image on a shot-by-shot basis. We have added the\nfollowing clarification in the revised manuscript:\nModified to (Addition):\n● Methods section p. 11 line 548 – 552:\n○ “To obtain a nominal background signal, we recorded images without gas or\nwater flow and averaged over 10 shots to improve the signal-to-noise ratio. The\nbackground was then subtracted from each signal image on a shot-by-shot basis.”\n2.7 Question 7: Usually, phase-contrast effects can be clearly seen on lineout profiles in the\nradiograph. Could the authors draw and show some profiles, e.g. for the data shown in Fig. 2?\nResponse:\nAs requested, we include below representative lineout profiles extracted from the experimental\nand synthetic phase-contrast images for different time delays. These profiles clearly highlight the\nedge enhancement and interference fringes characteristic of phase-contrast imaging. Additionally\na new Figure 9 was added/modified, which now incorporates previous Figure 2b along with a\nlineout comparison between experimental and simulated phase-contrast images, as well as their\ncorresponding density simulation.\nWe also appreciate the reviewer’s comment, which led us to revisit and refine our spatial\ncalibration. In doing so, we identified a misalignment in the assumed system magnification that\ndirectly affects the spatial scale (µm/pixel) and, by extension, the shock velocity. In the original\nmanuscript, we had assumed a nominal source-to-object distance of R = 0.2 m. However, during\n0\nthe process of aligning and overlapping the lineouts, we found that a more accurate value is R =\n0\n0.16 m. After this correction the simulation and experimental lineouts exhibit much better spatial\nagreement, as illustrated in the revised Figure 9.\nOriginal:\nModified to:\nOriginal:\n● Methods section 4.1 p. 11 line 537 – 539:\n○ …source-to-object distance R = 20 cm…\n0\n● Figure 3:\nModified to (Addition):\n● Methods section p. 11 line 525 – 526:\n○ …source-to-object distance R = 16 cm…\n0\n● Figure 3:\n● New Figure 9 has been added/modified:\n2.8 Question 8: What was the effective repetition rate in the experiment and how many shots\ndoes it take to have a full image of the dynamics? Besides, I would not say that 1 Hz is a high\nrepetition rate for such a system (it would have been for a real ICF experiment). In ref. 37 for\ninstance, the repetition rate was in practice 0.3 Hz - i.e., the same order of magnitude - for the\ncase of a solid target. Thus I suggest to remove or soften the mentions about the high-repetition\nrate operation.\nResponse:\nWe appreciate the reviewer’s suggestion and agree that the term ‘high-repetition-rate’ should be\nused with care, particularly when comparing different experimental platforms. As the reviewer\nnotes, in the context of inertial confinement fusion (ICF) experiments, repetition rates remain\n−5 −4\nextremely limited — often on the order of a few shots per day, 10 − 10 Hz. In contrast,\nHz. Our setup operates somewhere between these regimes, both producing high-energy-density\nconditions like in ICF, while enabling X-ray imaging with characteristics more akin to XFEL or\nsynchrotron-based diagnostics.\nAs described throughout the manuscript, the effective repetition rate of our experiment was 1 Hz,\nand the key dynamics were captured in single-shot exposures. In principle, concerning the\nimaging probe having up to femtosecond resolution. While 1 Hz may not be considered\nhigh-repetition depending on the context, we believe it does represent a significant advance over\nearlier high-intensity laser experiments as highlighted by Reviewer #1 , which for the past\n−1 −4\ndecade often operated at ~10 − 10 Hz. In our case, the repetition rate imaging the shock\ndynamics reflects the synchronized operation of both the laser wakefield accelerator (short-pulse\nlaser) and the shock-driving beam (long-pulse laser).\nImportantly, the liquid target system being used in our platform is, in principle, able of\n3 4\nsupporting even higher repetition rates for up to ~10 − 10 Hz, based on the fluid velocity to\nreplenish it, limited mainly by the laser and system engineering being used.\nThat said, we agree that 1 Hz repetition rate may not be considered high in all experimental\ncontexts. We have therefore revised the manuscript language as shown below,ff\nOriginal:\n● Introduction section: Line 119 – 123\n○ “Moreover, previous LWFA experiments have been restricted to single-shot, solid\ntargets, and many have not been able to exploit the high-repetition-rate laser\ncapabilities to image time-dependent dynamic systems.”\n● Results section: Line 149 – 151\n○ “To enable high repetition-rate operation, a free-flowing liquid (water) target was\ndesigned that is capable of providing a bulk plasma (30 µm diameter stream) in\nvacuum.”\n● Introduction section: Line 105 – 108:\n○ “The HTT laser system (λ = 800 nm) consists on a main LWFA beam line with\n0\na linearly-polarized (1.5 ± 0.2 J), high-repetition-rate (1 Hz)...”\nModified to:\n● Introduction section p. 3 line 117 – 122:\n○ “Moreover, previous LWFA experiments have been restricted to single-shot, solid\ntargets, and many have not been able to exploit the potential of\nhigh-repetition-rate laser capabilities in combination with liquid targets to image\ntime-dependent dynamic systems.”\n● Results section p. 3 line 148 – 151:\n○ “In combination, a free-flowing liquid (water) target was designed that is capable\nof providing a bulk plasma (30 µm diameter stream) in vacuum.”\n● Results section p. 3 ine 103 – 105:\n○ “The HTT laser system (λ = 800 nm, 1 Hz repetition rate) consists on a main\n2.9 Question 9: There is no laser filter in the setup depicted in Fig. 1. Where was it located?\nResponse:\nWe thank the reviewer for this helpful observation. While not explicitly illustrated in Fig. 1, the\nexperimental setup included a combination of protective laser-blocking and debris-shielding\nfilters placed in front of the in-vacuum X-ray CCD detector. Specifically, the filter stack\nconsisted of a 60 µm Aluminum foil, a 25 µm Kapton film, and a 36 µm Mylar layer. These were\nselected to attenuate residual laser light, reduce plasma self-emission, and protect the detector\nfrom debris.\nWe have added the following information to the Methods section for clarity,\nModified to (Addition):\n● Methods section p. 11 line 553 – 561:\n○ “To protect the in-vacuum CCD detector, and further reduce background signals,\na filter assembly was placed downstream from the detector. This consisted of a 60\nµm Aluminum foil, a 25 µm Kapton film, and a 36 µm Mylar layer, which\ntogether served to attenuate residual laser light, suppress plasma self-emission,\nand shield the detector from debris.”\n2.10 Question 10: For the phase contrast imaging application in the field of HED and ICF\nexperiments, it would be interesting to mention and discuss a different platform composed of a\nnanosecond X-ray backlighter coupled with a streaked camera. Such setup produces nice results\nat NIF - see e.g. Physical Review Letters 129.21 (2022): 215003, for the tracking of fuel-ablator\ninterface trajectory as well as the shock evolution inside the ablator in ICF implosions. Given the\nrelatively long (nanosecond) dynamics and time scale of such physical processes, is it really\nrelevant to use a femtosecond source, while an X-ray streak camera can provide <100 ps time\nresolution - and a gated full-frame detector around ~1 ns or less?\nResponse:\nWe appreciate the reviewer’s comparison. Indeed, nanosecond X-ray backlighters combined with\nstreak cameras or gated detectors are widely used in high-energy-density (HED) experiments for\ntracking slower hydrodynamic phenomena. In our experiment, the femtosecond duration of the\nbetatron X-ray source is not strictly required to resolve the nanosecond-scale shock dynamics.\nHowever, this ultrashort pulse duration is intrinsic to the nature of the laser wakefield accelerator\n(LWFA) and comes “for free” with the generation of betatron radiation. That is, we do not\nactively choose the femtosecond scale — rather, it is a byproduct of the electron beam\nacceleration and betatron emission process.\nWhile current large-scale facilities may not yet support fs-duration backlighters with sufficient\nbrightness, coherence, or repetition rate, the betatron source used here achieves all three\nsimultaneously in a compact geometry. Moreover, recent studies (e.g., Angulo et al.) suggest that\n100–200 ps time resolution is highly desirable for diagnosing HED hydrodynamics — a scale at\nwhich many traditional backlighters with gating are challenged.\nUltimately, this work aims to demonstrate a complementary approach with unique advantages in\ncompactness, source brightness, and spatial coherence. We have made a few changes to the text\nas shown below:\nReferences:\n● Angulo, Adrianna M., et al. \"Design of a high-resolution Rayleigh-Taylor experiment\nwith the Crystal Backlighter Imager on the National Ignition Facility.\" Journal of\nInstrumentation 17.02 (2022): P02025.\nOriginal:\n● Introduction section p. 2 line 84 – 92:\n○ “High-energy laser facilities commonly irradiate metal, foam, or gas targets with\nUV laser beams to create X-ray sources. These sources, however, suffer from…”\n● Discussion section p. 9 line 458 – 461:\n○ “... fine time resolution, excellent absolute timing and synchronization with\nthemselves as well as other lasers.”\nModified to:\n● Introduction section p. 2 line 86 – 88:\n○ “While these sources coupled with streak cameras can capture many\nHED-relevant processes, they suffer from…”\n● Discussion section p. 10 line 492 – 493:\n○ “... intrinsic femtosecond time resolution, excellent absolute timing and\nsynchronization with themselves as well as other lasers.”\n2.11 Question 11: The electron beam has a very broad spectrum. Can the Authors comment on\nthe effect of this energy spread when probing electromagnetic fields in the sample? There should\nbe important chromatic effects in the focusing/defocusing trajectories of the e-beam through the\nsample. Can't it lead to misinterpretation of the results?\nResponse:\nWe thank the reviewer for this thoughtful and important question. Indeed, the broad energy\nspread of the electron beam introduces chromatic effects in the form of energy-dependent\ndeflections, which can influence the accuracy of field magnitude recovery. To address this, we\nevaluated the field recovery using a range of assumed beam energies—specifically comparing\nresults from the low-energy tail (~20 MeV), medium energy (~44 MeV), and peak energy (~150\nMeV) of the measured spectrum as shown below,\nWhile variations in the assumed electron beam energy lead to noticeable differences in the\nrecovered field magnitudes, we find that the topology and spatial distribution of the fields remain\nconsistent across the evaluated energy range. This robustness is illustrated in Figure 5b of the\nmanuscript, where the orange error bands reflect the uncertainty introduced by the energy spread.\nOur methodology also agrees with the model presented by Zhang et. al, Ref. [43]. cited in the\nmanuscript, where similarly relativistic electron beams were used to reconstruct wakefields,\nacknowledging that while energy spread in the probe increases the probe density perturbations,\nthe increment is still rather small for even large energy spreads. As in their work, we find that\nwhile chromatic effects limit precision in absolute field strength, they do not compromise our\ninterpretation of field geometry and orientation – features most relevant to our conclusions.\nFinally, although our optimized electron beam reaches energies around 150 MeV, capturing the\nfull field dynamics it required effectively “detuning” the beam to improve spatial coverage. To\nselect a representative energy for field reconstruction under these conditions, we analyzed\nmeasured spectra in comparable datasets using an expectation value method. The analysis\nyielded a representative energy of E = 44 ± 14 MeV. Thus for its better representation of the real\n0\nspectral content, we adopted E = 44 MeV for the field recovery analysis. As seen in the figure\n0\nabove, this choice, along with comparisons across the energy range, supports the robustness of\nour topological interpretation while accounting for magnitude uncertainty via the error bands\nshown in Figure 5b.\nReferences:\n● [42] Zhang, C.J., Hua, J.F., Xu, X.L., Li, F., Pai, C.-H., Wan, Y., Wu, Y.P., Gu, Y.Q.,\nMori, W.B., Joshi, C.: Capturing relativistic wakefield structures in plasmas using\nultrashort high-energy electrons as a probe. Scientific Reports 6(1), 29485 (2016)\nhttps://doi.org/10.1038/srep29485\nOriginal:\n● Results section p. 9, line 409 – 412:\n○ This trajectory-crossing effect makes the fields challenging to analyze\nquantitatively, but lower limits on their strengths may be placed using a field\nrecovery method (see Sec. 4)\n● Methods section p. 15, Line 740 – 747:\n○ However, in the regime where the deflection angles α are large, the fields\n𝑥,𝑦\nbecome too strong, leading to particle crossing along the trajectories. This results\nin the formation of caustics, and implies that the relationship between 𝑥 and 𝑥\n0\nbecomes nonlinear, and the relationship between 𝐼 and 𝐼 is no longer unique –\n0\nlimiting the applicability of the algorithm for inversion of radiographs.\nModified to (Addition):\n● Results section p. 9 Line 431–436:\n○ Although caustics complicate quantitatively analysis of the fields, lower bounds\non their strengths and key features of their topology may still be extracted using a\nfield recovery method (Sec. 4.8.4).\n● Methods section (addition) p. 12 line 607 – 610:\n○ For the specific analysis of field recovery from electron radiographs, a\nrepresentative beam energy of E = 44 MeV was selected. This lower energy\n0\nreflects operating conditions where the beam was intentionally detuned to\nincrease divergence and maximize field sensitivity across the imaging region.\nAdditionally, this choice allows chromatic deflection effects to be better\naccounted for when measuring field magnitudes.\n● Methods section p. 16/17 line 807 – 821:\n○ Under strong-field conditions, or in the presence of low-energy electrons, large\ndeflection angles α can lead to overlapping beam trajectories, and the formation\n𝑥,𝑦\nof caustics in the radiograph. This affects the one-to-one mapping between the\nsource and detector coordinates, making the relationship between 𝑥 and 𝑥\n0\nnonlinear and the correspondence between 𝐼 and 𝐼 non-unique. While the broad\n0\nenergy spectrum of the electron beam can introduce chromatic effects in the\ndeflection response, which challenges precise field magnitude recovery. While\nthese should not be large as described in Ref. [42]. Key features of the field\ntopology, its evolution in time, and approximate lower bounds on field strength\nmay still be robustly extracted using this method.\n2.12 Question 12: Aren't there some drawbacks to this dual probe? In particular, since the\ne-beam crosses the sample and probably its mechanical mount, isn't it an additional source of\nnoise on the X-ray detector?\nResponse:\nWe thank the reviewer for the valid question. While the dual-probe geometry necessarily requires\nthe electron beam to traverse the sample, which could, in principle, generate bremsstrahlung or\nsecondary radiation that contributes to background on the X-ray detector. Several aspects of our\nexperimental design help mitigate this issue.\nMost notably, the target medium is liquid water — which is a low-Z material that produces\nsignificantly less bremsstrahlung compared to solid or high-Z targets. This choice is, in fact,\nadvantageous for minimizing parasitic radiation from the electron beam–target interaction.\nAdditionally, the X-ray detector was placed at a significantly longer distance from the interaction\nregion and shielded by filter packs, further reducing any potential background from broad-angle\nsecondary radiation. As a result, we did not observe any appreciable contamination or elevated\nnoise levels in the X-ray signal attributable to bremsstrahlung or other effects from the electron\nbeam–water interaction.\nRegarding its drawbacks, we acknowledge that the dual-probe setup introduces certain trade-offs,\nsuch as spatial constraints for sample–detector positioning, and increased complexity in target\nengineering. Nonetheless, we believe these limitations are outweighed by the substantial benefits\nof coordinated, multi-particle diagnostics. Notes have been added to the revised manuscript to\nreflect these considerations.\nModified to (addition):\n● Results section p. 4 line 198 – 202:\n○ “... and the low atomic number significantly reduces bremsstrahlung background\nfrom electron–target interactions compared to high-Z materials.”\n● Discussion section p. 11 line 511 – 516:\n○ “That said, the dual-probe setup does involves certain trade-offs, including spatial\nconstraints on sample and detector placement, characterization of each probe, as\nwell as added complexity in target engineering.”\n2.13 Question 13: - line 131 and section 4.5: I would write \"gas target\" instead of \"plasma\ntarget\"\nResponse:\nWe thank the reviewer for the suggestion, the text in line 131 has been updated accordingly:\nOriginal:\n● Results section p. 3 line 131:\n○ “Once the short laser pulse travels through the plasma target…’’\n● Methods section 4.5 header:\n○ “Plasma target’’\nModified to:\n● Results section p. 3 line 128:\n○ “Once the short laser pulse travels through the gas target…”\n● Methods section 4.5 header:\n○ “Gas target”\n2.14 Question 14: line 239: what does justify a polytropic index value of 5/3?\nResponse:\nWe thank the reviewer for raising this question. The choice of a polytropic index γ = 5/3 is\nstandard for modeling high-energy-density (HED) systems in the strongly shocked,\nnon-radiative, and weakly coupled regime. This value corresponds to an ideal monatomic gas\nand is commonly used in hydrodynamic simulations of laser-driven shocks where the energy is\nprimarily partitioned into translational degrees of freedom and radiation losses are negligible.\nThis assumption is supported by precedent in the literature, including work by Miles et al. cited\nbelow, and discussions in R.P. Drake’s textbook High-Energy-Density Physics (Ref. 48), which\noutlines the conditions under which γ = 5/3 is appropriate. We have added a clarifying sentence\nin the revised manuscript as shown below,\nReferences:\n● Miles, Aaron R. \"The blast-wave-driven instability as a vehicle for understanding\nsupernova explosion structure.\" The Astrophysical Journal 696.1 (2009): 498.\nhttps://doi.org/10.1088/0004-637X/696/1/498\n● [48] Drake, R.P.: High-energy-density physics: Foundation of inertial fusion and\nexperimental astrophysics, 658 (2018) https://doi.org/10.1007/978-3-319-67711-8\nOriginal:\n● Results section p. 5 line 237 – 239:\n○ “...and the polytropic index value is assumed to be γ = 5/3.”\nModified to:\n● Results section p. 5 line 245 – 247:\n○ “...and the polytropic index value is assumed to be γ = 5/3. Such index is typical\nfor strongly shocked, weakly coupled HED systems where radiative effects are\nminimal [48].”\n2.15 Question 15: In Methods, eq. 5: could the Authors provide a reference?\nResponse:\nWe thank the reviewer for the suggestion. We have now cited the reference included in the\nmanuscript:\n[54] F. Barbato, et al., Scientific Reports 9(1), 1–11 (2019)\nhttps://doi.org/10.1038/s41598-019-55074-1\nOriginal:\n● Methods section p. 11 line 535\n○ “... and PSF represents the detector's point-spread function.’\nModified to:\n● Methods section p. 11 line 522 – 523:\n○ “.. and PSF represents the detector's point-spread function as described in Ref.\n[54].”\n2.16 Question 16: Section 4.8: what is the physical justification of using a radial and an elliptical\nfrequency mask?\nResponse:\nWe thank the reviewer for this question. The use of radial and elliptical frequency masks in the\nFourier domain was not guided by a particular physical reason, but rather by established image\nprocessing techniques aimed at improving the image. The radial component serves to uniformly\nattenuate low-frequency background noise across all spatial directions, while the elliptical mask\naddresses anisotropic or structured noise—particularly periodic artifacts that may arise from\nexperimental asymmetries, such as a non-circular point spread function or geometric distortions.\nThe combined mask was empirically tuned to suppress these artifacts and enhance the visibility\nof relevant features in the phase-contrast images. This approach is commonly employed in both\nphase-contrast and biomedical imaging to increase signal-to-noise ratio and improve the clarity\nof subtle structural details. We believe the implementation described in Section 4.8.1 provides\nsufficient detail for reproducibility.\nFor reference, the figure below compares the image before and after applying the masking\nprocess. The “original” data refers to the minimally processed raw image (including only\nbackground subtraction and hot pixel removal), while the masked version incorporates the\nFourier-space mask described in the manuscript.\n2.17 Question 17: Moreover, I noticed few typos:\n- line 314/315: \"in the of order\"\n- line 640: \"by by\"\n- in ref 9 : DOI is https://doi.org/10.1063/1.3485109\nResponse:\nWe thank the reviewer for noticing these typos. The text has been updated to correct these:\nOriginal:\n● Results section p. 7 line 314- 315:\n○ “... in the of order the diameter of the stream.’\n● Methods section p. 13 line 640:\n○ “...as given by by…’’\n● Reference [9]\n○ “DOI: https://doi.org/10.1063/1.348510”\nModified to:\n● Results section p. 7 line 334 – 335:\n○ “... in the order of the diameter of the stream..”\n● Methods section p. 14 line 664:\n○ “...as given by…’’\n● Reference [9]\n○ “DOI: https://doi.org/10.1063/1.3485109”\nReviewer #3 (Remarks to the Author):\nMulti-messenger dynamic imaging of laser-driven shocks in water using a plasma wakefield\naccelerator\nSummary and General Comments:\nThis paper presents data combining ultrafast betatron X-ray imaging and relativistic electron\nbeam radiography, both generated by a laser wakefield accelerator to investigate the dynamics of\nlaser-driven shocks in a water jet target. The authors demonstrate that this “multi-messenger”\napproach provides complementary insights into hydrodynamic shock formation and associated\nelectromagnetic field evolution, which are not accessible with a single probe diagnostic. The\nwork addresses important challenges in high-energy-density physics, especially in the context of\nhigh rep-rate plasma/target diagnosis.\nResponse:\nWe thank the reviewer for the supportive comments.\nBy utilizing both ultrafast X-ray imaging along with electron beam radiography, the study\ncaptures detailed shock dynamics and intricates field structures. Precise calibration of both\ndiagnostics ensures accurate quantitative analysis. Additionally, the use of FLASH simulations\nreinforces the experimental findings by replicating the shock morphology and clarifying the\nimpact of the surrounding vapor layer.\nOverall, this manuscript presents sound results, is well-written and merits publication in Nature\nCommunications.\nResponse:\nWe appreciate the reviewer’s positive remarks.\nSuggestions for Improvement:\n3.1 Question 1: The authors state the shock breakout occurs at ~ 2 ns, which aligns with the\nmeasured shock velocity and reverse shock appearance time. However, the arrow in Figure 2c\nindicates breakout at around 4.5 ns. Please correct this discrepancy. Additionally, Figure 2a has a\nsignificant gap in the image sequence between 0.82 ns and 4.5 ns. Are images available at\napproximately 2 ns? If so, they should be included. Similarly, for the image sequence in Figure 5.\nResponse:\nWe thank the reviewer for these helpful suggestions. The shock breakout indeed occurs at\napproximately 2 ns, as stated in the text and supported by the analysis. The arrow in Figure 2c\nwas inadvertently misplaced and has now been corrected to point to the appropriate time frame.\nAdditionally, to improve clarity and completeness of the image sequence, we have updated both\nFigure 2 and Figure 5 to include frames captured around 1-2 ns. This fills the temporal gap\nmissing and provides a more continuous visualization of the shock evolution. We thank the\nreviewer for this helpful suggestion.\nOriginal:\n● Figure 2\n● Figure 5\nModified to:\n● Figure 2\n● Figure 5\n3.2 Question 2:: Figure 5 presents measurements and simulations of shock velocity up to 1 ns.\nI recommend including data points within the 1-2 ns interval (the breakout time) if exist, to better\nillustrate the linear increase up to the breakout time.\nResponse:\nWe value the reviewer’s constructive feedback. Regarding the shock velocity analysis in Fig. 3,\nwe have updated it to include additional data points leading up to breakout. We hope this\naddition offers a clearer depiction of the shock front’s progression and supports the observation\nof near-linear velocity trends before breakout timeframes.\nRegarding the field dynamics in Fig. 5, we note that most of the relevant features probed by the\nelectron beam occur within the first nanosecond. Frames beyond ~2 ns contain relatively limited\nnew information about the electromagnetic field structure. However, we agree that including a\nframe within the 1–2 ns window improves the continuity of the data presentation. Accordingly,\nwe have added this frame in the revised versions of Fig. 2 and Fig. 5.\nOriginal:\n● Figure 3:\n● Figure 5a:\nModified to:\n● Figure 3:\n● Figure 5a:\n3.3 Question 3: The author states: “FLASH simulations accurately model the fluid dynamics but\ndo not include electric (E) and magnetic (B) fields nor treat hot electron populations.” p.12,\nl.581. Please expand the discussion about the impact of the hot electron populations in the\npresent study (mainly for the broader audience).\nResponse:\nWe thank the reviewer for this thoughtful suggestion. Indeed, hot electron populations play an\nimportant role in laser–plasma interactions, particularly in the early stages of energy deposition.\nThese high-energy electrons, typically generated in the underdense plasma corona, can propagate\nover extended distances and contribute to nonlocal heat transport, influencing both the ablation\nprocess, and shock generation dynamics. Additionally, their motion drives return currents and\ncharge separation, giving rise to strong electric and magnetic fields — effects that are\nparticularly relevant for utilizing the electron beam probe.\nTo better clarify these points for the broader audience, we have added a few sentences in the\nrevised manuscript discussing the physical implications of hot electrons and the limitations of\nradiation hydrodynamic models like FLASH, which do not account for these effects. We hope\nthis new addition helps better contextualize discrepancies between simulation and experiment\nand highlights the need for future simulation models.\nModified to (Addition):\n● Results section p. 10 line 465 – 474:\n○ “The absence of hot electron populations in FLASH simulations is particularly\nrelevant in light of our observations. Hot electrons generated during laser–plasma\ninteractions can travel significant distances, driving non-local heat transport and\ndepositing energy into the surrounding vapor layer. These findings underscore the\nneed for future simulations that incorporate kinetic effects alongside radiation\nhydrodynamics, as well as advanced diagnostic systems in HED experiments.”\n3.4 Question 4: Provide further information on the sensitivity and limitations of the field\nrecovery method, especially in regions with caustics or large deflection angles, to better show its\nrange of applicability. The author states: “However, in the rExegime where the deflection angles\nαₓ and αᵧ are large, the fields become too strong, leading to particle crossing along the\ntrajectories...– limiting the applicability of the algorithm for inversion of radiographs.” p.15,\nl.740 Could the authors expand on how these limitations affect the recovered field magnitudes\nand the overall applicability of the method?\nResponse:\nWe appreciate the reviewer’s request for clarification. The limitations of the field recovery\nmethod, particularly in the regime of strong fields and large deflection angles where particle\ncrossing occurs, arises from the concept of one-to-one mapping between source and detector\nwhen strong fields or low-energy particles cause overlapping trajectories. In such cases, the\nalgorithm does not result in a unique solution, and the inferred magnitudes may be\nunderestimated or ambiguous. The extent of this effect depends on the electron beam energy of\nthe probe, as described in Question 2.11.\nHowever, the goal of our inversion is not to extract precise absolute field magnitudes but rather\nto reveal the topology and spatial structure of the fields, and particularly its time evolution. To\nillustrate this magnitude ambiguity effect, the orange error bands in Figure 5b reflect the\nuncertainty in recovered field magnitudes arising from these non-unique inversion regions.\nImportantly, we find that the overall field geometry and sign structure remain robust, even in the\npresence of inversion ambiguity. We have added some notes in the revised manuscript for\nclarification.\nOriginal:\n● Results section: p. 9, line 409 – 412\n○ “This trajectory-crossing effect makes the fields challenging to analyze\nquantitatively, but lower limits on their strengths may be placed using a field\nrecovery method (see Sec. 4).”\n● Methods section: p. 15, Line 740 – 747\n○ “However, in the regime where the deflection angles α are large, the fields\n𝑥,𝑦\nbecome too strong, leading to particle crossing along the trajectories. This results\nin the formation of caustics, and implies that the relationship between 𝑥 and 𝑥\n0\nbecomes nonlinear, and the relationship between 𝐼 and 𝐼 is no longer unique –\n0\nlimiting the applicability of the algorithm for inversion of radiographs.”\nModified to (Addition):\n● Results section p. 9 Line 431–436:\n○ “Although caustics complicate quantitatively analysis of the fields, lower bounds\non their strengths and key features of their topology may still be extracted using a\nfield recovery method (Sec. 4.8.4).”\n● Methods section p. 16/17 line 807 – 821:\n○ “Under strong-field conditions, or in the presence of low-energy electrons, large\ndeflection angles α can lead to overlapping beam trajectories, and the formation\n𝑥,𝑦\nof caustics in the radiograph. This affects the one-to-one mapping between the\nsource and detector coordinates, making the relationship between 𝑥 and 𝑥\n0\nnonlinear and the correspondence between 𝐼 and 𝐼 non-unique. While the broad\n0\nenergy spectrum of the electron beam can introduce chromatic effects in the\ndeflection response, which challenges precise field magnitude recovery. While\nthese should not be large as described in Ref. [42]. Key features of the field\ntopology, its evolution in time, and approximate lower bounds on field strength\nmay still be robustly extracted using this method.”\n3.5 Question 5: Please elaborate on the mechanism by which the low-density vapor layer\ninfluences shock symmetry. Additionally, provide a brief summary of the parameters used in the\nFourier and Meijering filtering techniques for image processing. The author states: \"Including\nthe low-density layer in the simulations enhances thermal transport around the water column,\nresulting in more uniform heating of the jet's exterior and producing a cylindrical shock and\nsymmetric compression morphology closely matching the experimental observations\" (p. 6, line\n300).\nSpecifically, explain in greater detail how this low-density vapor layer contributes to uniform\nheating and the resulting shock symmetry. Furthermore, outline the parameters\nemployed in the Fourier and Meijering filtering of the images.\nResponse:\nWe thank the reviewer for the question and suggestions. While the role of the low-density vapor\nlayer in promoting shock symmetry is discussed in detail in the manuscript (see lines 299–320 on\nfinal paragraph of page 6 / first paragraph of page 7), we agree this point might warrant further\nelaboration and we have included an additional note in the manuscript for clarity. The key\nmechanism is enhanced lateral heat transport: the partially ionized vapor provides a low-opacity\npartially ionized medium where 1 keV thermal electrons can travel further distances comparable\nto the target diameter (~20 µm) before depositing energy. This enables more uniform surface\nheating around the water column, resulting in symmetric ablation pressure and a more\ncylindrically shaped shock. In contrast, simulations without the vapor layer exhibit more\ndirectional and asymmetric shock profiles due to localized energy deposition.\nRegarding the image processing methods, the custom Fourier filter uses a mask as described in\nSec. 4.8.1 to improve the signal-to-noise ratio and suppress undesired spatial artifacts defined.\nThe mask is defined following the expression provided in Eq. 6 of the Methods section with\nparameters: a = 0.5, b = 2.0, and N = 16.\nFor the Meijering filter, we used the standard implementation from the skimage.filters.meijering\nfunction to enhance neurite-like features (e.g., shock edges) in both experimental and synthetic\nimages. The default parameters used are: sigmas = None, alpha=None, black_ridges=True,\nmode=’reflect’, cval=0.0 These clarifications have been added to the corresponding sections of\nthe revised manuscript as described below.\nOriginal:\n● Results section: Line 291 – 298\n○ “More importantly for our discussion is the difference in compression\nmorphology, where the vapor blanket provides a low-density medium in which\nelectrons are able to transport energy faster around the surface of the target. This\ntranslates to a more balanced pressure and density profiles, resulting in a\ncylindrically symmetric shock compression process.”\nModified to (Addition):\n● Results section p. 7 line 309 – 314:\n○ “More importantly for our discussion is the difference in compression\nmorphology, where the partially ionized vapor blanket provides a \\hl{low-opacity,\nlow-density medium in which electrons are able to transport energy more\nefficiently around the surface of the target.”\n● Methods section 4.8.1 p. 14 line 681 – 683:\n○ “In this work, we selected a = 0.5, b = 2.0, and N = 16 to optimize the quality of\nthe processed experimental images.”\n● Methods section 4.8.3 p. 15 line 721 – 724:\n○ “The Meijering filter was applied to both experimental and simulation images\nusing its default parameters as given by scikit-image filters package.”\n3.6 Question 6: This isn't a comment, but rather a point of discussion. The Atwood number is\n3/5 suggesting that the development of Richtmyer-Meshkov and later Rayleigh-Taylor\ninstabilities are possible. Given typical growth rates for the experiment described, the instability\nshould be detectable in Figure 2's data within a few nanoseconds, with amplitudes reaching\nseveral microns (a few pixels in the images). If the resolution is sub-micron, these perturbations\nshould be visible, especially at later times. Have data been collected that demonstrate density\nperturbations consistent with the Richtmyer-Meshkov instability? A tooth-like pattern is\ndiscernible in the data. Does the data at ~ 8 ns show an expanded version of this pattern? Please\ncomment in the text.\nResponse:\nWe sincerely thank the reviewer for this insightful and technically astute observation. Indeed, the\nhigh Atwood number and shock dynamics in our experiment provide favorable conditions for the\ndevelopment of Richtmyer-Meshkov instabilities. We are pleased to confirm that at later time\ndelays — particularly near at t > 3 ns — we observe clear, repeatable signatures of these\ninstabilities in the X-ray radiographs. The amplitude of the perturbations is consistent with\ntheoretical expectations and falls within the several-micron scale described by the reviewer, well\nresolved by our sub-micron-resolution betatron X-ray imaging system.\nGiven the richness and repeatability of these features, a detailed analysis describing their\ncharacteristic physics and the evolution of the instabilities is underway and will be presented in a\ndedicated follow-up manuscript. While we prefer to refrain from including additional data in the\npresent work, a brief mention has been added to the text to acknowledge the observation and its\nrelevance.\nModified to (Addition):\n● Discussion section p. 10 line 483 – 490:\n○ At later times, the X-ray probe further captures complex dynamics consistent with\nthe onset of post-shock plasma instabilities, as expected by the experimental\nconditions. Some of these features are appreciated in Fig. 2c at t > 3 ns, as signals\nnot discerned in the simulation model. A detailed analysis of these features is\nunderway and will be presented in future work.\n3.7 Question 7: Again, this isn't a comment, but rather a point of discussion. While the\nexperiment's diagnostic value is clear to experts, it is also clear that such an experimental layout\nis difficult to be implemented in an existing laser-fusion facility such as for instance NIF. However,\nthis study offers significant value for future facility designs.\nResponse:\nWe appreciate the reviewer's thoughtful perspective. We agree that integrating a dual-probe\nLWFA-based diagnostic system into a large-scale laser-fusion experiment like NIF would present\nsignificant logistical and infrastructural challenges.\nOur aim is not to propose an immediate implementation at such facilities, but rather to\ndemonstrate how compact laser-plasma accelerator platforms can serve as testbeds for advanced\ndiagnostic concepts, particularly those involving simultaneous photon and charged-particle\nprobing. The insights gained here — for example, regarding shock generation, target\ncompression symmetry, and electromagnetic field evolution— highlight the diagnostic value that\nsuch techniques could offer if appropriately scaled or adapted in future facility designs.\nAdditionally, we believe that such laser confinement fusion facilities would inherently possess\nthe laser expertise needed to support such advancements. These results could help motivate the\ncontinued development of high-repetition-rate, multi-probe diagnostics and may inform the\nplanning of next-generation HED science platforms.\nOriginal:\n● Discussion section p. 10 line 471 – 491:\n○ “Even though with this configuration it might be difficult to model exact\ndirect-drive ICF conditions, the purpose of this work is to highlight the power of\nthe multi-messenger probes for studying the interaction with both photons and\nelectrons, and to set a path forward towards more elaborate high-repetition-rate\nconfigurations with liquid targets. It displays a technique that is pioneering from\nprevious work, and rather unique when compared to other radiation sources\navailable. These insights may be invaluable for understanding similar phenomena\nin other complex dynamic systems, and for guiding the design of laser-plasma\ninteraction and ICF diagnostics experiments in smaller-scale laser infrastructures.\nMoreover, the multi-messenger technique may also motivate the construction of\nultrashort pulse laser systems capable of LWFA diagnostics directed to large scale\nfusion facilities such as the National Ignition Facility, Laser MegaJoule or LLE\nOMEGA.”\nModified to:\n● Discussion section p. 11 line 417 – 427:\n○ “While this configuration may not directly replicate full-scale direct-drive ICF\nexperiments, the purpose of this work is to demonstrate the diagnostic power of\nLWFA based multi-messenger probes and to set the path forward for more\nsophisticated, high-repetition-rate platforms using liquid targets. These insights\nmay also motivate the design of future smaller-scale HED science experiments,\nwhich could one day be adapted to large-scale fusion facilities such as the\nNational Ignition Facility, Laser MegaJoule, or OMEGA.”\n3.8 Question 8: The current literature on betatron generation via lasers within this article is\nlimited. Additional references utilizing laser systems of comparable power should be included.\nSpecifically, I recommend the inclusion of at least the following two references, both of which\nfocus on optimizing betatron radiation generation via Laser Wakefield Acceleration (LWFA):\nGrigoriadis, A., Andrianaki, G., Tatarakis, M., Benis, E. P., & Papadogiannis, N. A. (2021).\nBetatron-type laser-plasma x-ray sources generated in multi-electron gas targets. Applied Physics\nLetters, 118(13).\"\nYe, H., Gu, Y., Zhang, X., Wang, S., Tan, F., Zhang, J., ... & Zhou, W. (2022). Fast optimization\nfor betatron radiation from laser wakefield acceleration based on Bayesian optimization. Results\nin Physics, 43, 106116.\nResponse:\nWe thank the reviewer for their suggestion. While we believe that the manuscript already\nincludes a representative set of references covering various aspects of betatron X-ray generation\nvia laser wakefield acceleration [Refs. 32–39], particularly in relation to their capabilities for\nhigh-resolution imaging, we acknowledge that the field is rapidly evolving with many valuable\ncontributions in the literature. In response, we have added the first of the suggested references\nfrom Grigoriadis et al., (2021) to the manuscript.\nReferences (Addition):\n● [26] Grigoriadis, A., Andrianaki, G., Tatarakis, M., Benis, E.P., Papadogiannis, N.A.:\nBetatron-type laser-plasma x-ray sources generated in multi-electron gas targets. Applied\nPhysics Letters 118(13) (2021) https://doi.org/10.1063/5.0046184\n3.9 Question 9: Minor Note on Language:\nThere is a typo “chromatogrpahy” should be “chromatography.” p.12, l.586\nResponse:\nWe thank the reviewer for noticing this typo. The text has been updated:\nOriginal:\n● Methods section p. 12 line 586:\n○ “...The water was delivered to the chamber by a High-performance liquid\nchromatogrpahy (HPLC) pump.’\nModified to:\n● Methods section p. 12 line 597:\n○ “... The water was delivered to the chamber by a High-performance liquid\nchromatography (HPLC) pump.”\nReviewer #4 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is\npart of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have received numbers of comments and demands from the referees that they treated very seriously. I appreciate their detailed answers an I found the quality of the paper fairly improved. The paper is now ready for publication.", "reviewer_2": "I am completely satisfied with the response of the Authors. Their answers to my comments are convincing and well detailed, and I think that is the case also for other Referees' comments. I thus recommend the manuscript for direct publication in Nature Communications."}, "author_response": ""}]

Decision letter