Time-resolved X-ray imaging of the current filamentation instability in solid-density plasmas

Domain: physics
Published: 2026-01-09
Editorial outcome: minor_revision

Abstract
The streaming of energetic charged particles can magnetize astrophysical and laboratory plasmas via the current filamentation instability. Despite its importance, the experimental characterization of this instability has remained a challenge. Here, we report an experiment combining a high-intensity optical laser with a high-brightness X-ray free electron laser that successfully images the instability in solid-density plasmas with 200 nm spatial and 50 fs temporal resolution. We characterize the development of μm-scale filamentary structures and their evolution over tens of picoseconds through a non-linear merging process. The measured plasma density modulations and long merging time reveal the critical importance of space-charge effects and ion motion on this electron-driven instability. Supporting theoretical analysis and kinetic simulations help distinguish the relative role of space-charge and resistive effects. Our findings indicate that magnetic fields on the order of 10 megagauss are produced, with important implications for transport and radiation emission of energetic particles in plasmas.

Introduction
It has long been known that the propagation of high-energy charged particles in plasmas drives kinetic instabilities1 that produce growing electrostatic and electromagnetic fields. These (collective) fields control plasma dynamics and their study is critical to a wide range of systems, from cosmic-ray transporting galaxies2 to the solar wind3 to fusion plasmas4. Of primary interest is the so-called Weibel5, or current filamentation instability6, which plays an important role in the self-magnetization of many astrophysical and laboratory settings. These include the amplification of magnetic fields in young supernova remnant shocks7–9 and gamma-ray bursts10–12, which shape particle acceleration and radiation emission from these environments. This instability is also seen as a key candidate for the generation of the seed fields13 required to explain the ubiquitous galactic magnetic fields via a dynamo model14.

The study of the current filamentation instability, however, is challenged by its rapid growth and short (kinetic) spatial scale, which prevents its direct identification and characterization from astrophysical observations. Over the past decades, this has motivated significant interest in controlled laboratory studies using various experimental configurations, including electron beams from conventional and laser-driven particle accelerators15,16, field-ionized gases17, and non-relativistic colliding plasma flows driven by long (nanosecond) laser pulses7,18. In addition, there has been a substantial effort in investigating this instability in high-intensity laser-solid interactions, which are at the core of numerous cross-field applications, from advanced inertial fusion schemes4,19,20 to compact plasma-based accelerators21,22. In this context, the instability is central to understanding energetic electron transport and energy deposition, plasma heating, and magnetic field generation.

The high flux of relativistic (hot) electrons produced by an intense laser in a solid-density plasma generates a mega-ampere current23 and drives a return current in the background plasma. The resulting counter-propagating electron populations become unstable to the current filamentation instability and generate a magnetic field4,24. The strong density and temperature asymmetry between the two electron populations pose significant challenges to theoretical analysis; most theoretical and numerical studies have focused on the simplest symmetric case, where the instability is purely electromagnetic11,12. The few existing studies of the asymmetric case suggest that space-charge effects can play an important role25–27. Resistive effects on the dense background plasma further complicate the description of the instability27. The competition between these effects in the linear phase of the instability remains unclear. The non-linear evolution—critical to understanding the dynamics of the resulting magnetic field and its implications for the transport of energetic particles—is even less understood, highlighting the need for experimental characterization.

Previous experiments have primarily relied on optical probing24,28, as well as laser-driven electron29–31 and proton radiography21,32–34 of the laser-solid interaction to diagnose the current filamentation instability; however, these studies were limited to probing either far from the interaction region24,34 or the low-density plasma outside the solid target28,32. This is because optical radiation cannot penetrate the dense plasma and charged particles will be deflected by the strong electromagnetic fields generated at the target surface, preventing or corrupting the measurement within the interaction region. Moreover, kinetic scale processes, such as those associated with the filamentation instability at solid density, are particularly challenging to investigate, as they require very high spatial resolution (typically, sub-μm). While laser-driven X-ray sources can penetrate the dense plasma, they cannot yet deliver the required spatial resolution, narrow bandwidth and high flux needed to characterize the growth of this instability35–37.

Here, we take advantage of the high-brightness and narrow-bandwidth Linac Coherent Light Source (LCLS) XFEL to overcome these limitations in an X-ray phase contrast microscopy configuration, enabling an unprecedented characterization of the filamentation instability in solid-density plasmas. The resulting high-resolution images show the clear development of filamentary density modulations over picosecond time scales and reveal, as we will show, that space-charge effects and ion motion play an important role in significantly slowing down the growth of the instability and that strong, 10 MG-level, magnetic fields are produced.

Results
Our experiment was conducted at the Matter in Extreme Conditions (MEC) end-station of the SLAC National Accelerator Laboratory, in an optical-pump X-ray-probe configuration, as shown in Fig. 1. The high-contrast MEC short-pulse laser (0.8 μm wavelength, 1 J energy, 150 fs pulse duration, P-polarization)38 is focused on +10 μm copper and +15 μm aluminum wire targets with a focal spot of ~6 μm, reaching an intensity of ≈1.4×1019 W cm−2. The high peak-brightness Linac Coherent Light Source (LCLS) X-ray laser (650 μJ energy, 50 fs pulse duration) probes the laser-target interaction and subsequent plasma dynamics at 80° w.r.t. the pump-laser axis in an X-ray phase contrast microscopy configuration (see “Methods”).

By varying the timing between the optical pump and X-ray probe lasers from Δt = 0.5 ps to 800 ps, we identify three distinct phases of the plasma evolution. At early times (≤40 ps), we observe the development of clear filamentary structures, whose kinetic scale and transverse modulation are consistent with the current filamentation instability. At larger delays, the hydrodynamic evolution is revealed, with the observation of a laser-driven shock wave traveling at ~85 km s−1, and the thermal expansion of the heated target region. In the following, we will focus on the study of filamentary structures and associated plasma instabilities observed within the target, corresponding to Δt ≤ 40 ps.

Upon interacting with the solid density target, the high-intensity optical laser produces a small fraction of very energetic (hot) electrons39,40, with density nh ≈ ncrit, that propagate in the resulting plasma of density ne0 ≫ ncrit, where ncrit is the critical density for the propagation of the laser in the plasma. As the hot electrons reach the rear side of the target, a small fraction escapes and establishes a strong electrostatic sheath field, while the majority is reflected and recirculates within the target41. The sheath field accelerates rear surface ions via the Target Normal Sheath Acceleration (TNSA) process42. The energy spectrum of these energetic ions is measured via a Thomson Parabola Spectrometer (TPS), and can be modeled as dN/dE ∝ exp(−2E/Th)/E, where E is the particle energy43 (see “Methods”). Following this model, we can infer the slope temperature Th of the hot electrons, which is measured to be Th = 400 ± 30 keV and 700 ± 40 keV for the aluminum and copper targets, respectively. This is in good agreement with laser-induced electron heating models for high-contrast laser-solid interactions39,40 that predict hot electron temperatures in the range Th = 500–900 keV for the MEC laser conditions (see “Methods”).

The hot electrons drive a return current in the target as they propagate. The resulting counter-propagating electron populations then trigger the current filamentation instability. The associated plasma density evolution is captured through a series of single-shot images. These snapshots provide a spatial resolution of ≈200 nm and a temporal resolution determined by the 50 fs X-ray pulse duration, as well as the 100 fs precision of setting the pump-probe delay. High sensitivity to density variations is achieved via the novel MEC X-ray Imager (MXI) diagnostic44 that combines beryllium Compound Refractive Lenses (CRLs)45 positioned behind the target with an X-ray Camera. Due to the chromaticity of the CRLs and the bandwidth of the X-ray laser, both phase and absorption effects contribute to the X-ray signal, resulting in a high-resolution image with strong sensitivity to phase contrast effects introduced by density gradients inside the target.

The resulting X-ray images, displayed in Fig. 2, show the formation of density filaments with an initial wavelength λ ≈ 1 μm, evolving from Δt = 1.5–25 ps in the copper wire targets. Similar images are obtained for the aluminum targets, exhibiting filaments with an initial wavelength comparable to those in copper. We note that the distribution of wavelength components in the 3D structure is conserved through its 2D projection46, as well as the imaging system (see “Methods”). The laser-irradiated target surface remains very sharp during the growth of the filaments; the hydrodynamic evolution of the front surface only starts to be noticeable between 10 and 20 ps after the laser irradiation (see Fig. 2). This, together with the high-contrast of the X-ray images, clearly indicates that the filaments are produced in the solid-density plasma.

In the case of purely electromagnetic perturbations, both counter-propagating electron populations would be deflected at the same rate, producing symmetric filaments of alternating current polarity and no modulations in the total density. Since the MXI diagnostic is predominantly sensitive to gradients in electron density, such filaments would not produce a measurable signal. The alternating regions of high and low X-ray transmission, instead, show an asymmetric electron density distribution between filaments with opposite polarities, demonstrating that space-charge effects and ion motion must play an important role in the development of the instability in solid-density plasmas. This asymmetry arises because the hot and background electrons, having vastly different densities and temperatures, are deflected at different rates in the self-generated magnetic fields25. The resulting electron density perturbations produce electrostatic fields that set in motion the ions transversely to maintain quasi-neutrality. The plasma filamentation thus involves both electrons and ions even if the instability is triggered by the laser-produced fast electrons.

The evolution of the filament wavelength, shown in Fig. 3, is measured using lineouts of the X-ray images taken at different pump-probe delays (see “Methods”). The wavelength increases up to 2 μm for copper and 3 μm for aluminum over ≈20 ps, as a result of the non-linear evolution of the current filamentation instability.

Current filaments of the same polarity attract each other via the magnetic force, giving rise to an inverse cascade as they merge and increase in size47,48. This merging process can be modeled by considering a distribution of identical filaments with an average initial current I0, radius R0, and separation of 2R0. The magnetic field associated with each filament is B0 = 2I0/(cr), with r the cylindrical radius. The resulting attractive force per unit length acting on a neighboring filament of the same polarity is dF/dl = −B0I0/c. Following ref. 47 the wavelength evolution λ(t) is dictated by the pair-wise self-similar merging dynamics of the filaments, given by

λ(t) = λ0(2t/τm + 1)1/2, (1)

with

τm ≈ 0.39(mn)1/2λ0/B0 (2)

the characteristic merging time of two neighboring filaments, λ0 the wavelength after saturation of the instability (end of linear phase), m the mass of the particles in the filament and n their number density.

It is clear that the merging of filaments in the asymmetric regime cannot involve only the electron dynamics, as typically considered in the purely electromagnetic case47. Instead, the mass density carried during the merging process is dominated by that of the ions mn ≈ m ini to maintain quasi-neutrality, even if the current is carried by the electrons. This considerably increases the merging time of the filaments in the non-linear phase by a factor [mi/(Zme)]1/2 where Z is the ionization level.

We find that this merging model can provide a good description of the observed wavelength growth, as shown in Fig. 3. From fitting Eq. (1) to the wavelength measurements, the extracted merging time is τm ≈ 11 ps and τm ≈ 8 ps for copper and aluminum, respectively. From Eq. (2) the inferred magnetic field associated with the filaments is B0 ≈ 10 MG for copper, and B0 ≈ 7 MG for aluminum (see “Methods”). Such strong magnetic fields are consistent with the Alfvén limit for the fast electron current49. Considering that for a hot electron temperature of a few hundred keV, the typical drift velocity in the laser propagation direction is vdh ≈ 0.5c, the electron Larmor radius for B0 = 10 MG is 1 μm, which corresponds to the filament wavelength measured in the experiment. Once such large magnetic fields are produced in the plasma, they will affect fast electron transport, helping confined them within the filaments.

At later times Δt > 20 ps, the filaments start to rapidly decay. Interestingly, this is comparable to the predicted stopping range of hot electrons in the dense plasma. For a hot electron temperature Th = 700 keV, and typical values for the background plasma temperature Te = 100 eV and density ne0 = 5.8×1023 cm−350,51 (assuming a copper target with Z = 7), the stopping distance is Ls ≈ 0.35 cm52. For vdh ≈ 0.5c, this corresponds to a stopping time of τs ≈ 23 ps. Future experiments combining X-ray imaging with X-ray Thomson Scattering or K-alpha probes could help confirm whether the life time of the filaments is indeed determined by the hot electron stopping time and how the generated magnetic fields affect the fast electron transport.

We note that at such late times the filament detection is further complicated by the propagation of the laser-driven shock into the target and the increased background noise level, as can be observed in Fig. 2. Additionally, the MXI diagnostic becomes decreasingly sensitive to the filaments as their electron density gradient reduces with the growing wavelength.

The experimental results reveal the importance of electrostatic effects and ion motion in the dynamics of the asymmetric current filamentation instability produced in solid-density plasmas. Furthermore, they suggest that strong magnetic fields are produced, capable of affecting fast electron transport and confinement. We now compare these findings with fully kinetic simulations and analytic theory to further elucidate the interplay between these effects in the solid density plasma.

Kinetic simulations and theoretical analysis. We performed simulations of the laser-plasma interaction with the OSIRIS 4.0 particle-in-cell (PIC) code53. The laser irradiation of the copper wire and subsequent plasma evolution are described fully kinetically, including binary Coulomb collisions while maintaining a realistic plasma density and ion to electron mass ratio. The two-dimensional simulation domain models a lateral view of the laser-solid interaction, with a box size of 40×40 μm, centered on the +10 μm copper wire (see “Methods”).

The laser parameters are chosen to match the experimental conditions. Due to the high computational expense of kinetic simulations we focus on the first 3 ps of the interaction, which allows us to investigate the linear phase of growth and saturation of the filamentation instability. In particular, we focus our analysis on the microphysics governing the filament growth observed in the experiment and on the interplay between resistive and space-charge effects.

We first address the role of space-charge effects and ion dynamics by modeling the interaction without Coulomb collisions. As the laser-driven hot electrons propagate and recirculate in the plasma, we clearly observe the development of density filaments with a growth rate Γ ≈ 0.5 ps−1 and a dominant wavelength of λ ≈ 0.12 μm, as seen in Fig. 4a. During the growth of the instability we measure a characteristic hot electron slope temperature of Th ≈ 400 keV along the laser propagation direction and 100 keV transversely, an average drift velocity vdh ≈ 0.3c, and a density nh ≈ 0.02ne0. The large difference in temperature and density between the hot electrons and the cold return current populations drives an asymmetric instability characterized by total density modulations with δn/n ≈ 5%, consistent with the experimental observations. We verified that the large hot electron temperature transverse to the laser propagation direction renders this population stable to the collisionless current filamentation instability4, further confirming the importance of the cold return current. When the same simulation is repeated with ion motion switched off—that is, ions are modeled as an infinitely massive charge-neutralizing background species—density modulations are no longer visible. This highlights the critical role of space-charge effects in the development of the large density perturbations observed in both the simulations and experiment.

The simulations results are supported by theoretical calculations of the dispersion relation25. Figure 4d shows a comparison of the collisionless theoretical growth rate with and without coupling to electrostatic modes for the relevant hot electron and target parameters (see “Methods”). Space-charge effects are found to reduce the instability growth rate by more than an order of magnitude in comparison to the purely electromagnetic case, leading to a predicted growth rate Γ ≈ 0.6 ps−1 and most unstable wavelength λ ≈ 0.2 μm, in agreement with the collisionless simulations.

We then assess the effects of target resistivity by comparing the simulation results without and with Coulomb collisions, as illustrated in Fig. 4a, b. We find that in the resistive case the growth rate is increased to Γ ≈ 0.8 ps−1 with the most unstable mode shifted towards larger wavelength, leading to λ ≈ 0.7 μm, consistent with the experimental observations (see Fig. 4d, green and red markers). This migration towards larger wavelength, associated with the increased heating of the background electrons in the resistive case27, is predicted by theoretical calculations for the purely electromagnetic resistive current filamentation instability, as shown in Fig. 4d. A transverse lineout of the electron and ion density is shown in Fig. 4c, taken 3 μm away from the target surface in the collisional simulation. The ions are found to filament together with the electrons and produce density modulations with a similar amplitude when compared to the collisionless simulation, demonstrating the importance of ion dynamics in both the resistive and collisionless cases. The filament growth is accompanied by strong amplification of the magnetic field reaching B ≈ 7 MG (Fig. 4c, black curve). This is again consistent with the Alfvén limit for the hot electron drift velocity vdh ≈ 0.3c and filament wavelength λ = 0.7 μm measured in the simulations, and in good agreement with the values inferred from the filament merging rate in the experiment.

Our results reveal the combined importance of space-charge and resistive effects on the growth and non-linear dynamics of the current filamentation instability, which significantly slow down its growth rate and increase the dominant wavelength. The strong, 10 MG-level magnetic fields produced by the instability can impact various applications, potentially enabling high brilliance synchrotron radiation sources54,55 and improving the collimation of fast electrons in fast ignition inertial confinement fusion schemes. More generally, this work opens up a new experimental route for detailed studies of the microphysics of relativistic streaming instabilities of relevance to both laboratory and astrophysical plasmas12,56. Future extensions to currently available higher laser intensity and lower density targets21 could probe deeper nonlinear regimes and the mechanisms behind the formation of relativistic shockwaves relevant to gamma-ray bursts10,12. Furthermore, at higher laser intensities, even larger magnetic fields will be produced, potentially enabling direct measurements of density and magnetic field structures by combining X-ray imaging with advances in X-ray polarimetry57.

Methods
Experimental setup
The experiment presented in this work was conducted at the SLAC National Accelerator Laboratory. At the Matter in Extreme Conditions (MEC) end station we combined the MEC high-intensity optical short-pulse laser with the Linac Coherent Light Source (LCLS) high peak-brightness X-ray laser in a pump-probe configuration. The 1 J, 150 fs, 800 nm Ti:Sapphire high intensity contrast (>107 at t = −3 ps)38 MEC pump laser (P-polarization) was focused on 10 μm copper and 15 μm aluminum wire targets via an f/6 Off-Axis Parabola (OAP). The focal spot is of 5 μm FWHM along the wire axis direction and 7 μm along the transverse direction, containing 60% of the laser energy and generating intensities on target of ≈1.4×1019 W cm−2. The 8 keV, 650 μJ, 50 fs LCLS X-ray probe laser was pre-focused to a beam diameter of +150 μm via upstream CRLs to back illuminate the laser-target interaction at an angle of 80° w.r.t. the optical pump laser.

The attenuated X-ray beam was imaged onto an Andor Neo 5.5 sCMOS camera via an additional set of CRLs placed after the target to achieve a spatial resolution of ≈200 nm. Due to the chromaticity of the beryllium CRLs and the bandwidth of the LCLS beam (~20 eV), the imaging system possesses a focal range instead of a single focal point. This leads to an effective propagation distance d = 0.8–1 mm for the X-rays, which ensures that both absorption and phase effects contribute to the obtained images. The MEC time tool allows for precise measurement of the delay in arrival between the pump and probe beams, with a range of 100 fs. A leakage of the optical laser upstream is used to image the wire target in three microscopy imaging systems to aid the alignment process and increase the shot rate. A Transmission diagnostic captures the transmitted light of the interaction to deliver a measure of the interaction quality. A Thomson Parabola Spectrometer and Ion Imager measure the accelerated ion beam energy spectrum and spatial profile. Additionally, the combination of these secondary diagnostics allows us to quantify the quality of the interaction on shot and deliver dismissal criteria for the X-ray images. Any measurements with a large laser transmission and simultaneous poor flux and ion cut-off energy measured on the Ion Imager and Thomson Parabola Spectrometer were discarded from the analysis.

Laser contrast and pre-plasma formation
The temporal contrast of the MEC Ti:Sapphire laser was characterized using a Tundra autocorrelator, showing excellent performance up to the main pulse38. The intensity contrast remains better than 10−9 up to −5 ps and better than 10−7 beyond that. To estimate the preplasma formation due to this contrast profile, we consider that significant laser energy absorption into solid copper begins above an intensity of ~1012 W cm−258. This threshold is reached approximately at −3 ps for the MEC laser contrast curve. By integrating the laser intensity from −3 to −0.5 ps and applying intensity-dependent absorption coefficients from the same reference, we estimate that ~2 μJ of energy is deposited in the target prior to the main pulse. Assuming the energy is deposited within a 25.8 nm thick surface layer59 over a ~5 μm diameter spot, and using copper’s atomic density with partial ionization (Z = 4–5), we estimate an electron temperature of 30–50 eV. This yields a plasma expansion velocity of ~0.02 μm ps−1. Given the short time between the onset of the pre-pulse and the arrival of the main pulse, the plasma does not expands significantly. As a result, the estimated preplasma scale length at the time of interaction remains below 0.1 μm, allowing the laser to interact with a steep density gradient near the original solid surface.

Ion energy spectrum and hot electron temperature
The Thomson parabola spectrometer was set at a 36° angle w.r.t. the laser axis to measure the spectrum of energetic ions generated during the interaction. A low divergence portion of the beam is extracted using a +130 μm pinhole at a distance of 1.315 m from the target, corresponding to an acceptance angle of 0.77 μsr. The ions are then deflected by a 0.525 T magnetic field to distinguish particles of different energy along one direction, and a 6 kV cm−1 electric field to differentiate particles with different charge-to-mass ratio along the other direction. The particles are visualized by a microchannel plate (MCP) with a phosphor screen and imaged with an OPAL-4000 CCD camera, providing a 2D image of continuous spectra of all the ions within the beam. The Thomson Parabola is calibrated in energy using a set of aluminum filter of different thickness, resulting in a minimum resolvable proton energy of ~0.6 MeV, which is constrained by the size of the MCP. The raw image in Fig. 1 shows traces corresponding to copper, carbon, oxygen ions and protons. These accelerated ions originate from atoms located on the wire surface, including common contaminants such as hydrocarbons or water vapor. The majority of ions in the beam consist of protons, attributed to the TNSA sheath field’s preferential acceleration of low charge-to-mass ratio ions. The resulting proton spectrum is fitted as

dN/dE = (c n t S sheath / E) exp(−2E/Th), (3)

with E the particle energy, t the acceleration time, c the sound speed and S sheath the electron sheath surface43. The slope temperature Th of the hot electrons on the rear surface can be inferred by fitting the proton spectra according to Eq. (3), as seen in Fig. 5. We obtain an average hot electron temperature of 400 ± 30 and 700 ± 40 keV with standard deviation of 200 and 260 keV for the aluminum and copper targets, respectively. The variability in slope electron temperature can be attributed mainly to the narrow acceptance angle and distance of the TPS to the wire targets.

The hot electron temperature extracted from the ion spectra can then be compared to previously established collisionless laser-heating models39,40. For relativistic, short-pulse, high-contrast laser-solid interactions, the hot electron temperature is primarily determined by the normalized laser vector potential, a0 ≈ 2.560,61. In this case, both the expressions found in Eq. (2) of Wilks et al.39 and Eq. (10) of Haines et al.40 give comparable hot electron temperatures, in the range of Th = 500–900 keV, consistent with the experimentally obtained values.

X-ray image processing
Due to defects and absorption effects in the beryllium CRL stacks, background removal techniques were used to reveal the density distribution inside the wire. Since the X-ray beam profile possesses a jitter in both pointing and intensity, the shot post-processing procedure relied on ≈200 images for both the X-ray background and the X-ray imaged cold (reference) wire target. The X-ray background is then removed on all shots, delivering a transmission map of the wire. An additional normalization of the shot image with a cold reference wire image Ishot/Iref is very effective in revealing changes in transmission induced by the density perturbations inside the target.

Filament wavelength measurements
Since the 2D X-ray images obtained from the MXI are a projection of a 3D forest of filaments evolving within the target, it is necessary to ensure that the filament wavelength is conserved through its projection and imaging system. This is shown in Fig. 6. Approximating the filament density distribution with a sinusoidal signal, the wavelength conservation through the 3D structure can be explained via the linearity of the Fourier transform, meaning that a sum of multiple sinusoidal signals conserve the Fourier components of the individual signals. Since the accumulation of both absorption and phase contrast effects along the X-ray propagation axis can be described as a sum of the effects caused by individual sinusoidal filament rows, the wavelength components of the filament forest is conserved by the 2D projection.

Verifying the conservation of the filament wavelength through the imaging system requires modeling the imaging system synthetically to understand its effects on the obtained images. The phase effects are produced by the chromaticity of the beryllium CRLs, leading to a range of foci45, which produce an effective propagation distance d between the target and the imaging plane. This effective propagation distance is determined experimentally from undriven cold wire reference images. Analytically, we model the MXI as a perfect thin lens imaging system with a focusing error ϵ = 1/f – 1/do – 1/di, where f, do, and di are the focal length, object distance, and imaging distance.

Any material then modulates an incoming plane wave such that the function f(x,y), describing the target plane before propagation, can be written as

f(x,y) = exp[i 2π T(x,y) / λ] = exp[−i δT(x,y) 2π/λ] exp[−βT(x,y) 2π/λ], (4)

where T(x,y) is the two-dimensional thickness function of the object that the X-rays of wavelength λ interact with, and δ, β are the real and imaginary part of the refractive index.

After passing through the target, the modulated plane wave in the input plane propagates in free space for a propagation distance d = d0 − di ≈ ϵd2 to an output plane g(x,y). The output function can then be described as the product of the transfer function Ψ(νx,νy) = exp[iπϵd2λ(νx2+νy2)] with the Fourier transform of the input function in Fourier space62

g(x,y) = Ψ ∫∫ F(νx,νy) exp[−iπλd(νx2+νy2)] exp[−i2π(νx x + νy y)] dνx dνy, (5)

with Ψ = exp(id2π/λ). Since the propagated wave attained from g(x,y) is subsequently imaged on the detector, a synthetic X-ray image can be calculated via I(x,y) = |g(x,y)|2. We can extract the full Fresnel diffraction pattern on our detector produced by a thickness mask representing the targets in the experiment. To emulate the limitation in the resolution of the imaging system, a Gaussian filter is applied to the synthetic images to artificially limit the resolution to ≈200 nm.

The synthetic X-ray images are used to systematically analyze the response of the imaging system to sinusoidal density perturbations. Due to the strong sensitivity to density gradients the imaging system appears to distort the signal in the case of very strong density perturbations concentrated in small filaments. However, even in such scenarios the overall wavelength, meaning the distance between same polarity filaments, remains conserved.

The filament wavelength is extracted via 1D lineouts taken at various distances from the target front surface. A Fourier filter is applied to remove the high and low frequency components of the signal as seen in Fig. 7. The resulting signal exhibits a range of different wavelengths and amplitudes clearly distinguishable from the background. Each maximum and minimum, corresponding to filaments of opposite polarity, is individually fitted by a Gaussian distribution function. To account for the susceptibility of the imaging system to extreme electron density gradients, filaments are organized in pairs of opposite polarity. The distance between each filament pair is then used to extract a distribution of filament wavelengths for each shot and distance from the front surface.

Filament merging model
The merging of filaments after the saturation of the instability can be modeled using a simplified approach, following ref. 47. The main steps of the derivation are reproduced here for completeness. We consider a distribution of straight, internally uniform filaments with identical current, polarity, mass density and radius, separated by a distance D0 = λ0/2, and interacting pair-wise via the attractive force per unit length, dF/dL = −B0I0/c. Here B0 = 2I0/(cr) is the magnetic field associated to a filament and r is the distance to the axis. Interactions involving more than two filaments or filaments of opposite polarities are neglected. The equation of motion becomes

d2x/dt2 = −2I02 / (c2μ0x), (6)

where μ0 is the lineic mass. In the asymmetric regime, the mass density carried during the merging process must be dominated by that of the ions mn ≈ m ini to maintain quasi-neutrality even if the current is carried by the electrons, such that μ0 = nmi(πD02/4). The initial merging time, that is, the time for two filaments to cross the distance D0, may be evaluated analytically from Eq. (6) as

τm0 = 2I0 / (c D0) [π μ0 / 2]1/2 ≈ 0.39(mn)1/2 λ0 / B0. (7)

The pair-wise merging process occurs in a self-similar way, since each step of filaments merging gives rise to identical starting conditions but with rescaled parameters. After the kth merging step, the current and lineic mass of the filaments evolve as Ik = 2kI0 and μk = 2kμ0, and the separation distance from their neighbors increases as Dk = 2k/2D0 such that λk = 2k/2λ0. The time required to reach the kth step is tk = (2k/2 − 1)τm0. The resulting wavelength evolution may then be calculated as

λ(t) = λ0(2t/τm0 + 1)1/2, (8)

and can be used to infer the magnetic field if the merging time and initial wavelength are known.

By fitting this model to the experimental measurements shown in Fig. 3, the merging time is τm = 11.0 ± 0.4 ps for copper and 8.53 ± 0.25 ps for aluminum. Using Eq. (2) the inferred magnetic field is B0 = 10.64 ± 0.13 MG and B0 = 7.8 ± 0.1 MG for copper and aluminum, respectively. Here, the uncertainties are derived from standard error propagation based on the determination of the wavelength, the noise level of the background, as well as fitting errors and do not account for the simplifying assumptions of the model.

Particle-in-cell simulations
The two-dimensional simulations of the laser-target interaction have been performed for the +10 μm copper wire with the relativistic and massively parallel PIC code OSIRIS 4.053. The laser irradiation of the solid density target and subsequent plasma evolution are described fully kinetically, using a realistic ion-to-electron mass ratio and realistic plasma density ni = 8.49×1022 cm−3. The copper ions are considered to be ionized 7 times, based on the results of a 0D three-temperatures model of the laser-target interaction50. The PIC simulations include Coulomb collisions between all populations using a relativistic Monte-Carlo operator63.

The 2D simulation domain models a slice of the wire with a box size Lx (wire radius direction) = Ly (wire axis direction) = 40 μm and spatial resolution of 6.9×10−3 μm. The boundary conditions for both particles and fields are open along the x direction and periodic along the y direction. The interaction is followed for several picoseconds with a timestep Δt = 0.011 fs to ensure proper calculation of the collisions and avoid numerical heating. The P-polarized laser pulse with a wavelength of 0.8 μm and normalized laser potential a0 = 3 is focused on the wire front surface. It follows a Gaussian temporal profile in intensity of 150 fs FWHM and a Gaussian spatial profile with 6 μm FWHM transversely to the target normal. The pre-plasma is modeled assuming an exponentially decreasing density profile with a scale length of 0.07 μm, which is consistent with the laser contrast and the particles from solid density, with n0 = 0.1 ncrit. We have used improved numerical accuracy when depositing the current on the grid. We have tested the convergence of the results by varying the spatial resolution (0.5–1 c/ωpe), the timestep (0.49–0.71 ωpe−1), the number of particles per cell per species (64–400), the box size (Lx = 30–40 μm, Ly = 40–80 μm) and the pre-plasma scale length (0.03–0.12 μm). We also performed a 2D simulation with a circular target (top-view) and found that it did not significantly affect the fast electron properties and the filamentary structures that develop in the dense plasma.

Dispersion relation of the current filamentation instability
The growth rate and most unstable wavelength of the current filamentation instability can be calculated from the dispersion relation derived for a system of two drifting Maxwell–Boltzmann electron distributions:

f0s(v) = ns / (2πvths2) exp[−(vx − vds)2 + vy2 / (2vths2)], (9)

where the index s = h, c corresponds to either the hot and return current populations and vths = (Ts/ms)1/2 is the thermal velocity. We consider the electron counter-streaming velocity vds to be in the e^x direction and, without loss of generality, the wavenumber of the unstable filamentation mode to be k = ky e^y. The choice of non-relativistic distributions is justified by the dominant role of the return current, which is non-relativistic, and the relative insensitivity to the hot electron temperature in the conditions of the experiments as we will discuss below. The return-current electron population provides current neutralization and a cold non-drifting background ion distribution provides charge neutralization Zni = ne0.

Collisionless filamentation. Following the standard linearization of the Vlasov-Maxwell equations, considering the contributions of both electromagnetic (EEM = E e^x) and electrostatic (EES = E e^y) fields, and perturbations of the type /ei(kyy−ωt) the collisionless dispersion relation is given by25,26

det[ω2 − ωpi2 − c2ky2 0; 0 ω2 − ωpi2] [Σs (ωps2/ω2) (ζs2 Z0s + ζs Zs + ζs2 Z0s0) / (ζs Zs + ζs2 Z0s0 − 3/2 Z0s0 − 4/1 Z0s00)] = 0, (10)

where Zs(ξs) = 1/π1/2 ∫−11 dt e−t2 / (t−ξs) (11)

is the plasma dispersion function with ξs = ω/(2ky vths)1/2, ζs = vds/(2vths)1/2 and the number of primes in the Z functions denotes the number of differentiations. The two diagonal terms are responsible for the electromagnetic and electrostatic modes, whereas the off-diagonal terms give rise to the coupling between both modes.

Reduced expressions for the fastest growing mode can be obtained in the case of purely electromagnetic perturbations, i.e., neglecting the coupling to electrostatic modes. Considering ω = iΓ with Γ the growth rate of the filamentation instability, and taking the relevant limits for the conditions of the experiment and simulations α = nh/n0 ≪ 1, vthc ≪ vdh and vthc ≪ vthh (corresponding to the kinetic limit), the dispersion relation becomes insensitive to the hot electron temperature and is given by

Γ2 + (1−α) (π/4)1/2 (α2 βdh2 / βthc) (Γ2 + (1−α) βdh2) / (1−αβdh2) + k2 = 0. (12)

In Eq. (12) and hereafter the growth rate Γ is normalized by the electron plasma frequency ωpe = (4πne0 e2/me)1/2, the wavenumber k is normalized by ωpe/c and β = v/c is the normalized velocity. The wavenumber subscript has been omitted to simplify the notation. To leading order in k2βthc2 ≪ 1 the collisionless dispersion relation becomes simply

Γ = (2kβdh / π1/2) (α2 βdh2 / (1−α) βthc) (1−αβdh2) − k2. (13)

The maximum growth rate is

Γmax = (2/3) (α3 βdh3 / 3π(1−α)5/2 βthc2) (14)

and corresponds to the most unstable wavenumber

kmax = (α βdh / (3π(1−α))1/2 βthc). (15)

In this regime, the properties of the return current control the growth of the instability with a growth rate that is much smaller than the plasma frequency.

By numerically solving the full collisionless dispersion relation of Eq. (10), we find that space-charge effects play an important role in this regime further decreasing the growth rate. The results for the conditions informed by PIC simulations, namely n0e = 5.8×1023 cm−3, α = 0.02, βdh = 0.3, βthc = 0.45, Te = 100 eV, Z = 7 and me/mi = 1.158×105, are shown in Fig. 4d.

Resistive filamentation. The importance of collisional effects in the dense plasma regions of solid-density target or ICF targets have led to significant efforts in studying the impact of collisions on the filamentation instability. Given the much lower density of the hot electrons with respect to the background in these scenarios, collisions affect primarily the non-relativistic return current population.

Most previous work has explored the impact of collisions by using the BGK operator26,27. These works found that collisions could either enhance or mitigate the filamentation instability, depending on the regime. The simplicity of the BGK operator is significant, but it is limited at large growth rates and relativistic times, on top of additional drawbacks related for example to the non-conservation of particle number.

In the case of the current experimental and simulation conditions, the collision frequency is very large. For example, for a copper plasma with n0e = 5.8×1023 cm−3, Z = 7 and Te = 100 eV the electron collision frequency is νe ≈ 0.16 ωpe63, which is much larger than the growth rate of the collisionless filamentation instability (Γ ≈ 4×10−5 ωpe).

In order to analyse the role of collisional effects in this regime, we consider a resistive model of the filamentation instability containing the same two counter-streaming electron distributions given by Eq. (9) and a charge neutralizing ion population, but where the electric field is now given by a simple Ohm’s law E = ηJ with η the electrical resistivity. Following the linearization of the Vlasov-Maxwell equations, the resulting dispersion relation for electromagnetic fluctuations is given by34

Γ2 + (1−α) (Γ / νe) (βdh2 / βthh2) (1 + (π/2)1/2 (Γ / kβthh) exp(Γ2 / 2k2βthh2) (1−erf(Γ / 2kβthh))) + k2 = 0. (16)

To leading order in Γ/(kβthh) ≪ 1 and νe2 αβdh2 / βthh2 ≪ 1 the growth rate is

Γ = (αβdh2 / (1−νe2 α + (π/2)1/2 (kβthh / αβdh2))) − k2. (17)

The comparison of the full resistive (Eq. (16)) and collisionless (Eq. (10)) dispersion relations is shown in Fig. 4d. We find that in the regime of interest for the experiments and simulations, the main role of collisional effects is to shift the most unstable wavelength towards larger scales.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the manuscript \"Time-resolved X-ray imaging of the current filamentation instability in solid density plasmas\" authors present their recent experiment on the current filamentation in the laser plasma, where the X-ray free electron laser (XFEL) was used for probing the developing density modulations. The study is relevant for such fields of research as astrophysics, inertial confinement fusion, laser plasma ion acceleration, etc. Direct experimental characterization of the current filamentation instability in the solid density laser plasmas is very challenging as it requires penetration of such media by the probe. In the earlier studies, the laser-plasma accelerated protons were considered to the probe the fields within plasma, as they have sufficiently short duration and proper mass-to-charge ratio for this task. The original feature of the present study and is the use of XFEL pulses along with the multi-TW laser that can drastically improve the temporal and spatial resolution in measuring directly the plasma density. Currently, the access to such joined XFEL and laser-plasma experiments is very limited, which puts this work in rather a unique position.\nThe experiments had the single-angle pump-probe setup with a wire target, and the measurements included the X-ray phase-contrast plasma imaging, and the ion spectra measurements with the Thomson parabola. The plasma density filaments were observed over the first tens of picoseconds after the laser, and were analyzed in terms of their amplitude and spatial frequency. Comparing these observables with the simple theoretical and 2D PIC simulations, allowed estimating the electron plasma temperature and magnetic fields, as well as determining the role of such processes as ion motion and collisions in the instability development and termination.\nThe paper is well written and explains the material comprehensively. The presented experimental results are new and they demonstrate how XFEL pulses can be used to probe fine structures in solid plasmas. In my opinion, the paper may be published in Nature Communications, but before recommending it I would like to suggest following remarks:\n- the estimates of electron temperature and magnetic field are provided with high numerical accuracy (e.g. B_0=10.64±0.13) and rather the modest errorbars supposedly obtained from fitting with theoretical models (e.g. Eqs (1,2), PIC modeling). To me it feels not very rigorous, as all these models are based on strong simplifications (reduced geometry, pre-ionized plasma, identical filement currents etc), so they cannot physically provide such an accuracy, and this may be misleading as it give the impression that this method can indeed be used to measure magnetic field inside the plasma with 1% precision. The validity of the estimates should be discussed thoroughly and without any implicit exaggerations.\n- line 122: authors mention that electron temperature deduced from ion spectra agree with heating models. it would be instructive provide the used expressions for the temperature and the conditions under which it was applied;\n- lines 177 and 186 : authors provide the estimate of electron current velocity of `0.5*c` but I've failed to find where this number is coming from -- intuitively it makes sense, but it should be defined by the interaction of the laser with the pre-plasma and plasma skin-layer and may vary depending on the actual heating mechanism.\nAfter authors address these comments I would be able to recommend this manuscript for the publication.", "reviewer_2": "The paper “Time-resolved X-ray imaging of the current filamentation instability in solid density plasmas” by Schoenwaelder et al. describes the use on high resolution x-ray imaging to diagnose the current filamentation instability in solid density copper plasmas.\nHowever I am not convinced that the authors' explanation for these measurements is justified.\nThe spatial resolution of the diagnostic is impressive, and it shows filamentary structures apparently expanding into the 5-micron wire. The filaments must be electron and ion density modulations to be measurable – and since they persist over > 10 of picoseconds. (100 x greater than the laser pulse duration!) However, it seems strange that hot electron populations are trapped in a filament for time duration of this length. There have been many previous measurements of the duration of x-ray pulses from such experiments (K-alpha etc) and the time scale is much less. Other work has shown the hot electrons refluxing throughout the plasma and propagating very far from the interaction region.\nThe experimental work is accompanied by PIC simulations which show some similar behavior on shorter timescales, shorter spatial scales and in 2D while also using a very small initial density scalelength.\nThe data is very nice but there has also been a lot of work previously on this instability in HED plasmas– not all of which has been referenced.\n1)Experimental Study of Current Filamentation Instability\nB. Allen et al.,Phys. Rev. Lett. 109, 185007 (2012)\n2) Observations of the filamentation of high-intensity laser-produced electron beams\nM. S. Wei et al., Phys. Rev. E 70, 056412 (2004)\n3) Weibel-Induced Filamentation during an Ultrafast Laser-Driven Plasma Expansion\nK. Quinn et al., Phys. Rev. Lett. 108, 135001 (2012)\n4) Dynamics of the Electromagnetic Fields Induced by Fast Electron Propagation in Near-Solid-Density Media L. Romagnani et al. Phys. Rev. Lett. 122, 025001 (2019).\n5) Current filamentation instability in laser wakefield accelerators\nC M Huntington et al., Phys Rev Lett., 106 :105001 (2011)\nQuestions:\n1) Why are the simulations not 3D? It seems that the cylindrical geometry of the wire will play a significant effect on the physics – especially since the wire diameter is about the same as the focal spot diameter.\n2) Why do the filaments only extend to the center of the wire? This would be expected if the density modulations observed are due to plasma structures on the outside surface of the wire (i.e. the area that has been irradiated). How do the authors know that the density modulations are not due to such hydrodynamics structure (perhaps induced by pre-pulses) on the irradiated surface of the wire.\n3) The experimental electron temperature is derived from the slope of the proton spectrum. This is problematic since the proton spectra is highly dependent on the laser pulse duration (total pulse energy) rather than the peak intensity. There are many review papers about this.\n4) What is the laser contrast and the laser and the expected (or measured) density scalength?", "reviewer_3": "The Ms \"Time-resolved X-ray imaging of the current filamentation instability in solid density plasmas\" reports about imaging laser-matter interaction with very high time resolution using the combination of a FEL and probing laser-matter interaction. The topic of radiography is always an interesting topic within the community as far as it can unlock some bottlenecks within this research area. As such, the paper could be interesting for the community. However, probing of matter using X-rays has already been done in the past, such as using laser-accelerated protons to resolve quick electromagnetic phenomena (e.g. see the PRL by Romagnani 2005). For the moment it is therefore not completely clear, what superiority the quite complex setup used by the author has compared to what has been used in the past and what additional information can be retrieved with this complex setup in addition to what other methods would be able to detect. I think that authors should emphasize this in their Ms such as to convey a clearer message and show superiority of their (quite complex) setup compared to \"easier\" solutions using only X-rays or laser-accelerated protons.\nMore in detail:\nAbstract: authors write \"Our findings indicate that magnetic fields on the order of 10 megagauss are produced, which will help confine high-energy electrons within the filaments with important implications for the transport of energetic particles in plasmas.\"\nWhat about electric fields ? Are they not of importance ? What implications can this have ? What applications have authors in mind that could have an impact on this ?\nPage 3: the resolution seems to be 50 fs and <200 nm. Again, what would be the resolution using laser-accelerated protons (that can probe at ps time frames) or laser-accelerated X-rays, is a resolution of 200 nm required ?\nPage 5: it is not completely clear what \"The temporal evolution of the filament wavelength\" is indicating ? How has this information been retrieved from Fig. 2 ? Maybe a small explanation (arrows etc) could help the reader understand the underlying calculations.\nPage 7: the statement \"These experimental results reveal a critical importance of electrostatic effects and ion motion in the dynamics of the asymmetric current filamentation instability\" - how is this importance revealed ? The ion motion seems to be a new information, how is that information retrieved ?\nPage 8 (and theory): it would be interesting to compare the PIC results to the experimental results. Authors state that there is a good agreement in the text, but finally there is no figure comparing the results, which would strengthen their statement, even visually.\nPage 9: Author mention: \"Our results reveal the combined importance of space-charge and resistive effects on the growth and nonlinear dynamics of the current filamentation instability.\" - What does this mean ? Are the two, space-charge and resistive effects, individually important, or only its combination ? What is the impact on the filamentation individually.\nIn general, the main message of the paper is not fully clear: Initially it seems that the main message is about a new imaging diagnostics (\"Time-resolved X-ray imaging\". Then it evolves to the importance of magnetic fields in plasma and finally to filamentation issues, space-charge and resistive effects. I would encourage the authors to find a guideline (\"fil conducteur\") related to what they want to say within the Ms and build the content/results/discussion accordingly."}, "author_response": "We thank the Reviewers for carefully reading our manuscript and for their constructive\ncomments, which have helped us improve our manuscript. Below, we reproduce the Reviewers\nreport (in blue), provide a detailed point-by-point response to each comment (in black) and\nindicate the corresponding changes in the manuscript. All changes made in the manuscript are\ncolor-highlighted in the revised version.\nWe are confident these changes have helped clarify the importance and novelty of our results\non the characterization of a fundamental plasma instability in high-energy-density conditions,\nwhich has long been a significant experimental challenge and is relevant to a wide range of\nsystems and applications, from high-energy astrophysics to the energy transport in inertial\nfusion plasmas to the development of laser-driven compact secondary sources. We hope that\nthis work can now be accepted for publication in Nature Communications.\nReviewer #1:\nIn the manuscript \"Time-resolved X-ray imaging of the current filamentation instability in solid\ndensity plasmas\" authors present their recent experiment on the current filamentation in the\nlaser plasma, where the X-ray free electron laser (XFEL) was used for probing the developing\ndensity modulations. The study is relevant for such fields of research as astrophysics, inertial\nconfinement fusion, laser plasma ion acceleration, etc. Direct experimental characterization of\nthe current filamentation instability in the solid density laser plasmas is very challenging as it\nrequires penetration of such media by the probe. In the earlier studies, the laser-plasma\naccelerated protons were considered to the probe the fields within plasma, as they have\nsufficiently short duration and proper mass-to-charge ratio for this task. The original feature of\nthe present study and is the use of XFEL pulses along with the multi-TW laser that can\ndrastically improve the temporal and spatial resolution in measuring directly the plasma density.\nCurrently, the access to such joined XFEL and laser-plasma experiments is very limited, which\nputs this work in rather a unique position.\nThe experiments had the single-angle pump-probe setup with a wire target, and the\nmeasurements included the X-ray phase-contrast plasma imaging, and the ion spectra\nmeasurements with the Thomson parabola. The plasma density filaments were observed over\nthe first tens of picoseconds after the laser, and were analyzed in terms of their amplitude and\nspatial frequency. Comparing these observables with the simple theoretical and 2D PIC\nsimulations, allowed estimating the electron plasma temperature and magnetic fields, as well as\ndetermining the role of such processes as ion motion and collisions in the instability\ndevelopment and termination.\nThe paper is well written and explains the material comprehensively. The presented\nexperimental results are new and they demonstrate how XFEL pulses can be used to probe fine\nstructures in solid plasmas. In my opinion, the paper may be published in Nature\nCommunications, but before recommending it I would like to suggest following remarks:\nWe would like to thank the Reviewer for highlighting the novelty and broad importance of our\nstudy as well as the unique characterization of the filamentation instability enabled by it, which\n1\nhas long been a significant experimental challenge. Below, we address, point-by-point, the\nReviewer’s remarks which have helped us improve our work.\nThe estimates of electron temperature and magnetic field are provided with high numerical\naccuracy (e.g. B_0=10.64±0.13) and rather the modest errorbars supposedly obtained from\nfitting with theoretical models (e.g. Eqs (1,2), PIC modeling). To me it feels not very rigorous, as\nall these models are based on strong simplifications (reduced geometry, pre-ionized plasma,\nidentical filement currents etc), so they cannot physically provide such an accuracy, and this\nmay be misleading as it give the impression that this method can indeed be used to measure\nmagnetic field inside the plasma with 1% precision. The validity of the estimates should be\ndiscussed thoroughly and without any implicit exaggerations.\nThe values for the precision of the inferred magnetic field are obtained from standard error\npropagation based on the determination of the wavelength, the noise level in the X-ray images,\nas well as fitting errors of the theoretical model to the experimental data. The Reviewer is\ncorrect that they do not account for the underlying simplifying assumptions of the model and we\nagree that this should be clarified.\nWe have revised the main text to give the approximate values of the magnetic field and make\nclear how these are obtained from fitting to the experimental measurements (lines 189 - 193 in\nrevised manuscript). We also added a more detailed discussion to the methods (section:\nFilament merging model, lines 456-462 in revised manuscript), where we include the error bars\nfrom the procedure above, and explain that they do not account for the simplifying assumptions\nof the underlying model.\nline 122: authors mention that electron temperature deduced from ion spectra agree with\nheating models. it would be instructive provide the used expressions for the temperature and\nthe conditions under which it was applied;\nThe expressions used correspond to Eq. (10) and Eq. (11) of [Haines, M. G. et al., Physical\nReview Letters 102, 045008 (2009)], with Eq. (11) describing the pondermotive scaling as found\nby [Wilks, S. C. et al., Physical Review Letters 69, 1383–1386 (1992)]. These models work\nunder the assumption of relativistic, short-pulse, laser-solid interactions, which is the relevant\nregime in our MEC experiments with a high-contrast laser and a laser vector potential a0 = 2.5\n[Malka, G. & Miquel, Physical Review letters 77, 75-78 (1996); Rusby et al. Physics of Plasmas\n31, 040503 (2024)]. In this regime T_hot is primarily determined by the normalized laser vector\npotential. Both models give comparable temperatures in the range T_hot = 500 - 900 keV,\nconsistent with the values inferred from the measured ion spectrum.\nWe have modified the sentence addressing the heating models in the main text to state that\nthese models apply for high-contrast laser-plasma interactions (lines 127 - 130 in revised\nmanuscript). We have also added a more detailed discussion in the methods section (“Ion\nenergy spectrum and hot electron temperature”, lines 364 - 370 in revised manuscript), that\n2\nincludes how the hot electron temperature is extracted from the ion spectra, as well as the\nreferences to the hot electron temperature models and their underlying assumptions.\nlines 177 and 186 : authors provide the estimate of electron current velocity of `0.5*c` but I've\nfailed to find where this number is coming from -- intuitively it makes sense, but it should be\ndefined by the interaction of the laser with the pre-plasma and plasma skin-layer and may vary\ndepending on the actual heating mechanism.\nIndeed, the details of the electron distribution function do depend on the laser and plasma\nproperties. However, for relativistically intense lasers and solid-density (overdense) targets, the\naverage energy of these electrons can be estimated based on the slope temperature of the\nlaser-accelerated electron energy distribution, which scales primarily with laser intensity (or\nnormalized vector potential) as discussed above. For the conditions of our experiments T_hot is\na few hundred keV. Based on this, it is reasonable to estimate that a typical velocity of the fast\nelectrons in the forward direction is a large fraction of c, and we have thus considered a value of\n0.5c. This is of course a simple estimate for the typical electron drift velocity and is in\nreasonable agreement with the average fast electron drift velocity measured in the simulations\nv=0.3c.\nWe have revised the text in the manuscript to make this clearer (lines 195 - 198 in revised\nmanuscript).\nReviewer #2:\nThe paper “Time-resolved X-ray imaging of the current filamentation instability in solid density\nplasmas” by Schoenwaelder et al. describes the use on high resolution x-ray imaging to\ndiagnose the current filamentation instability in solid density copper plasmas.\nHowever I am not convinced that the authors' explanation for these measurements is justified.\nThe spatial resolution of the diagnostic is impressive, and it shows filamentary structures\napparently expanding into the 5-micron wire. The filaments must be electron and ion\ndensity modulations to be measurable – and since they persist over > 10 of picoseconds.\n(100 x greater than the laser pulse duration!) However, it seems strange that hot electron\npopulations are trapped in a filament for time duration of this length. There have been many previous\nmeasurements of the duration of x-ray pulses from such experiments (K-alpha etc) and the\ntime scale is much less. Other work has shown the hot electrons refluxing throughout the plasma\nand propagating very far from the interaction region.\nWe thank the Reviewer for the careful consideration of our work and for recognizing the\nimpressive nature of our measurements. We are happy to further clarify the importance of these\nmeasurements and their interpretation.\ndensity modulations in the plasma and must persist for >10 ps, as discussed in the manuscript.\nThis, in itself, is already a very important result and observation of our work. As we will discuss\nbelow, it is also clear that these modulations occur inside the dense plasma, thus constituting a\nvery important and long-sought characterization of the density evolution of the filamentation\ninstability in high-energy-density plasmas.\nThe Reviewer is correct that, on these time scales, the hot electrons can recirculate within the\ntarget, although this does not prevent them from driving a return current that is unstable to the\ncurrent filamentation instability. In the absence of strong magnetic fields, if electrons move\nballistically, an important fraction will leave the central (interaction) region along the wire axis\ndue to their divergence, and the number of hot electrons in the interaction region will decrease\nover time, which can affect the long-time dynamics of the instability. We do not measure the fast\nelectron dynamics and lifetime directly in the experiment. However, if strong magnetic fields are\nproduced with magnitudes on the order of the Alfvén limit, as inferred from the experimental\nanalysis and observed in the corresponding simulations, then, as we argue in the paper, this\ncould significantly affect their transport along the axis of the wire, as these fields are\nperpendicular to it. For example, while the electrons will continue to recirculate in the target and\nsome fraction can still escape from the central region, once such strong fields are established\nthey would help confine the electrons and significantly limit their escape. In this case, one would\nexpect their lifetime to be primarily determined by collisions within the plasma as they recirculate\nin the target. As indicated in the manuscript, the collisional time for 700 keV electrons is ~20 ps,\nconsistent with the duration of the density modulations observed experimentally. At the same\ntime, we note that it is the return current electrons that drive the instability and the relative\nimportance of these two populations for the lifetime of the filaments is not yet well established,\nneither theoretically nor experimentally.\nWe agree that complementing our measurements with simultaneous K-alpha or X-ray Thomson\nscattering diagnostics would be very interesting and in fact we have just submitted a proposal to\ndo that. However, we would like to mention a couple of points regarding the relation between\nK-alpha, X-ray emission, and the lifetime of fast electrons in these type of experiments:\n1) We are not sure which references the Reviewer is referring to. It is correct that there have\nbeen publications indicating sub-picosecond K-alpha durations [e.g. Feurer et al. Appl Phys B\n72, 15–20 (2001)]. However, there have also been several experiments using ~150 fs duration\nlasers indicating K-alpha emission for durations of > 10 ps (much longer than the laser) [e.g.\nAndiel et al. Phys. Rev. E 63, 026407 (2001); Bastiani-Ceccotti et al. Appl Phys B 78, 905–909\n(2004); Shepherd et al. UCRL-TR-229579, 908084 (2007); Nagel et al. Appl. Phys. Lett. 110,\n144102 (2017); Sawada et al. Nature Communications 15, 7528 (2024)];\n2) If K-alpha emission originates primarily from fast electrons in the target, one could expect a\nshorter signal associated with the burst of fast electrons that are produced during the\nlaser-pulse duration, followed by a longer duration signal associated with the refluxing of these\nelectrons in the target and their collisional equilibration process. The relative strength of these\n4\ntwo emission phases will depend on the details of the interaction, the target size, and the\ngeneration of fields inside the plasma. Moreover, the ability to clearly detect and separate them\nwill depend on the sensitivity of the diagnostics and on their temporal resolution;\n3) To our knowledge, a clear characterization of these phases using K-alpha has not been\nestablished in these types of experiments and as such the connection between K-alpha\ndiagnostics and the presence/properties of fast electrons needs to be more carefully\nunderstood.\nWe have revised the manuscript to indicate that the life time of the fast electrons in the target is\nnot well established and that it would be interesting to have simultaneous X-ray Thomson\nscattering or K-alpha diagnostics in the future to better characterize the fast electron properties\nin connection to the lifetime of the filaments (lines 205 - 208 in revised manuscript).\nThe experimental work is accompanied by PIC simulations which show some similar behavior\non shorter timescales, shorter spatial scales and in 2D while also using a very small initial\ndensity scalelength.\nIndeed, fully-kinetic PIC simulations including realistic target density and collisions have been\nperformed. They capture the development of the instability produced by laser-driven electrons\nduring the first 3 ps, indicating good agreement with the experimental data on the spatial and\ntemporal scales of the filamentary density structures and helping to elucidate the interplay\nbetween electromagnetic, space-charge and collisional effects on the growth of the instability.\nDue to the tremendous computational expense, the simulations are limited to 2D and a few ps.\nWe will further elaborate on how prohibitive 3D simulations are and why the density scalelength\nused is appropriate in our answers to the comments below.\nThe data is very nice but there has also been a lot of work previously on this instability in HED\nplasmas– not all of which has been referenced.\n1) Experimental Study of Current Filamentation Instability\nB. Allen et al.,Phys. Rev. Lett. 109, 185007 (2012)\n2) Observations of the filamentation of high-intensity laser-produced electron beams\nM. S. Wei et al., Phys. Rev. E 70, 056412 (2004)\n3) Weibel-Induced Filamentation during an Ultrafast Laser-Driven Plasma Expansion\nK. Quinn et al., Phys. Rev. Lett. 108, 135001 (2012)\n4) Dynamics of the Electromagnetic Fields Induced by Fast Electron Propagation in\nNear-Solid-Density Media L. Romagnani et al. Phys. Rev. Lett. 122, 025001 (2019).\n5) Current filamentation instability in laser wakefield accelerators\nC M Huntington et al., Phys Rev Lett., 106 :105001 (2011)\nWe thank the Reviewer for acknowledging the quality of the data presented in the manuscript\nand for the references on previous studies of filamentation in laser-plasma interactions. We\nwould like to highlight that our experiment is the first to directly image and resolve the\n5\nspatio-temporal evolution of the density modulations associated with this instability in\nsolid-density plasmas. Following the Reviewer’s suggestion, we have now added the additional\nreferences to the manuscript.\nWhy are the simulations not 3D? It seems that the cylindrical geometry of the wire will play a\nsignificant effect on the physics – especially since the wire diameter is about the same as the\nfocal spot diameter.\nUnfortunately, it is not possible to model the interaction in 3D for several ps while taking into\naccount the correct solid plasma density, collisions, and the laser finite spot size/focusing\neffects. The 2D simulations of the laser-wire interaction presented in the manuscript already\nrequired ~1 million CPU hours each, modeling ~10 billion particles. A 3D simulation would\nrequire ~1 billion CPU hours, which is completely out of reach even for the largest\nsupercomputers.\nDespite these computational constraints, we have performed different simulations in order to\nassess the role of the target cylindrical geometry in the instability formation. In particular, we\nhave performed additional 2D top-view (round-target) simulations with the same target and laser\nparameters as in the side-view case presented in the manuscript. The results are shown in Fig.\n1 of this response below. As we can see, the growth and spatial scale of the density filaments,\nas well as the peak magnetic field amplification, is very similar to that observed in the planar\ngeometry, with the density perturbation amplitude being ~70% of the value observed in the\nplanar case. In the manuscript, we opted to include the simulation analysis based on the 2D\nside-view geometry, as that corresponds to the probing geometry in the experiment. We also\nnote that the extent of the filaments inside the target (up to ~5 μm for the copper target) is\nindependent of the target geometry. This will be further discussed in our answer to the next\ncomment.\nWe have added a statement in the Methods regarding these results (“Particle-In-Cell\nSimulations section”, lines 484 - 489 in the revised manuscript).\nc) Lineouts of the electron density (blue), ion density (red) and magnetic field intensity (black)\ntaken at x=-3μm for the top-view simulation and d) for the side-view simulation. The data is\ntaken at t=1.1 ps during the growth of the filaments.\nWhy do the filaments only extend to the center of the wire? This would be expected if the\ndensity modulations observed are due to plasma structures on the outside surface of the wire\n(i.e. the area that has been irradiated). How do the authors know that the density modulations\nare not due to such hydrodynamics structure (perhaps induced by pre-pulses) on the irradiated\nsurface of the wire.\nThe longitudinal extent of the density filaments is indeed a very interesting question that we are\nactively exploring. We have carefully checked that this is not associated with the target\ngeometry or hydrodynamic structures on the outside of the target surface.\nThe MEC short-pulse laser possesses very good contrast, with pre-pulses only carrying ~ 2μJ\n(see additional details in the answer to the following comments). This is largely insufficient to\nproduce any significant hydrodynamic modulations of the target before the main pulse arrives.\nwith the cold wire reference images (see Fig. 2 of this response), which show that the target\nfront surface is not modulated and remains sharp for multiple picoseconds. Additionally, we also\nsee that the hydrodynamic evolution of the target surface from the main pulse interaction occurs\non much longer time scales at t > 20 ps with the propagation of a shock and following the\nthermal expansion of the target [as observed in Fig. 1 of the manuscript]. As a result,\nhydrodynamic modulations cannot be responsible for the filamentary structures observed.\nFigure 2: X-ray images of (a) the cold (static) wire reference and (b) the laser-driven wire 1.5ps\nafter the interaction. (c) shows the division between the laser-driven image and the cold\nreference image to reveal the X-ray transmission relative to the static cold wire target (as seen\nfor multiple pump-probe delays in Figure 2 of the manuscript). As can be seen in these images,\nthe front of the target is still sharp at 1.5ps after the laser-target interaction. The edges of the\ntarget are visible in c) due to flux variation in the X-ray beam and resulting changes in the phase\ncontrast effects of the very steep density front, leading to slight variations of the fresnel\ninterference pattern in these near-field images. (d) shows the division between two different cold\nreference images taken of the same wire target that show the same edge fringes to confirm that\nthe edges in (c) are not caused by the laser-target interaction.\nIt is also important to point out that our X-ray images are most sensitive to the solid density\nplasmas and sharp gradients therein. Such strong contrasts in density could not be produced in\nan expanded pre-plasma outside the wire, where the density would be too low for X-rays and\nthe associated density gradients would need to be much sharper to produce similar\nimages—not consistent with hydrodynamic origin.\nWe also observe that the extent of the filaments can be longer than the wire center and reach\nclose to the full diameter as shown in Fig. 3 of this response for the case of the Al wire with a\ndiameter of 15 micrometer, thus indicating that their extent is not determined by the irradiated\narea. We note that the extent of the filaments observed in the experiments is also consistent\nwith that observed in our simulations. We have been exploring the simulations and theory\nfurther in order to develop a model of how the fast electron density and divergence as well as\n8\nthe target collisionality control the spatial extent of the filaments that will be left for a separate\npublication.\nFigure 3: X-ray image of a 15μm aluminum target at t = 20ps. The filaments clearly develop past\nthe center of the wire target.\nWe have added a section to the Methods addressing the laser-contrast and expected\npre-plasma scale length (“Laser contrast and pre-plasma formation, line 321 - 338 in revised\nmanuscript). We have also added a sentence in the main text of the revised manuscript that\nmentions that the hydrodynamic evolution is evolving on longer time scales and cannot explain\nthe observed density filaments (Lines 148 - 152 in revised manuscript).\nThe experimental electron temperature is derived from the slope of the proton spectrum. This is\nproblematic since the proton spectra is highly dependent on the laser pulse duration (total pulse\nenergy) rather than the peak intensity. There are many review papers about this.\nThe Reviewer is correct that for long pulse (> ps) lasers and significant pre-plasmas the proton\nspectrum can depend strongly on the pulse duration and on the fast electron production in an\nextended pre-plasma. However, this is not the regime of our experiment.\nAs we discuss in detail in the next point, in our experiment the level of pre-plasma was low\ngiven that the laser pulse intensity contrast was very good: ~10⁻⁹ until –5 ps and ~10⁻⁷ after\n(see Fig. 4 in our answer to next comment). The total energy in the pre-pulse is of the order of\n~2µJ. Given that the pre-pulse arrives only a few picoseconds before the main pulse, there is\ninsufficient time for significant hydrodynamic expansion, and thus the pre-plasma scale length is\nexpected to remain well below 1 µm. Under similar conditions, previous studies, such as by\nMalka and Miquel (Phys. Rev. Lett. 77, 75, 1996), as well as recent review papers, for example\nRusby et al., Phys. Plasmas 31, 040503 (2024), support the ponderomotive scaling for\nestimating the hot electron temperature, due to the negligible role of pre-plasma formation.\nplasma expansion to occur during the laser-plasma interaction itself. Indeed, we see that our\nmeasured proton spectra are in very good agreement with Mora’s model for a\nnon-pre-expanded plasma slab (see extended Figure E1 of the manuscript). In order to develop\nfurther confidence in our estimate of the fast electron temperature, we compare the derivation\nfrom the slope of the proton spectrum with the main theoretical models that have been\npreviously established in the literature (namely, the Wilks’ and Haines’ models), finding good\nagreement as well. Finally, the fast electron temperature is also in good agreement with the PIC\nsimulations.\nFor all these reasons, we are very confident in the fast electron temperature that is derived from\nthe experimental measurements and corresponding theoretical models.\nWe have revised the manuscript to explicitly mention and discuss the high-contrast of the MEC\nlaser (lines 99 and 127 in revised manuscript) as well as in the Methods section (“Laser contrast\nand pre-plasma formation, lines 296 and 321 - 338 in revised manuscript), and added a more\ndetailed discussion on the models for the fast electron temperature and their regime of validity in\nthe Methods (lines 364 - 370 in revised manuscript).\nWhat is the laser contrast and the laser and the expected (or measured) density scalength?\nThe MEC short-pulse laser achieves high temporal contrast through a pulse cleaning technique\nbased on noncollinear sum-frequency mixing of the signal and idler from an optical parametric\namplifier (OPA), as described in [E. Cunningham et al., Appl. Phys. Lett. 114, 221106 (2019)].\nAs shown in the Tundra autocorrelation trace (Fig. 4 of this response), the contrast reaches\n~10⁻⁹ up to –5 ps and ~10⁻⁷ beyond that. A feature at ~3 ps is noted but, as mentioned in the\npublication, it is likely an artifact of the contrast measurement diagnostic. In the following, we\ninclude a back-of-the-envelope calculation of the expected pre-plasma scalelength due to this\npre-pulse.\nUsing absorption data from femtosecond laser interactions with solid copper [S. E. Kirkwood et\nal., Phys. Rev. B 79, 144120 (2009)], significant energy absorption begins around an intensity\nthreshold of ~1×10¹² W/cm². This increase corresponds to changes in electron surface\ntemperature and electron-electron collision frequency. According to the MEC contrast curve, this\nintensity threshold is crossed at approximately –3 ps. By integrating the laser intensity from –3\nps to –0.5 ps and applying intensity-dependent absorption coefficients for copper from [S. E.\nKirkwood et al., Phys. Rev. B 79, 144120 (2009)] (~5% for I <1012 W/cm2, ~10% for 1012< I <\nL L\nL\nduring this pre-pulse window. Beyond –0.5 ps, the contrast measurement is limited by the\ndiagnostic instrument function.\nAssuming this energy is deposited and retained within a surface layer defined by a 5 µm\ndiameter and a depth of ~25.8 nm (10.8 nm from optical penetration plus ~15 nm from ballistic\n10\nhot electron transport [Rethfeld et al., J. Phys. D 50, 193001 (2017)]), we estimate an affected\nvolume of V = 5.07×10-13 cm3. Using solid copper’s atomic density (n = 8.45×1022 atoms/cm3),\nCu\nthe number of atoms in the volume is N ≈ n ×V = 4.28×1010. Assuming full deposition of 2 µJ\ni Cu\n(E ) and partial ionization (Z ≈ 4–5), an upper estimate for the resulting free electron\nL\ntemperature is T = 2/3×E [eV]/(Z×N ) = 30 - 50 eV. Calculating the resulting plasma expansion\ne L i\nvelocity, we obtain v = (ZkT /m )1/2 ~ 0.02 µm/ps (Z = 5, Te = 50 eV). Since the pre-pulse\nexp e Cu\narrives only a few picoseconds before the main pulse, the resulting pre-plasma does not have\ntime to significantly expand, and the scalelength remains below 0.1 µm at the time of interaction.\nBased on these calculations, we have also explored the role of pre-plasma in the interaction\nusing PIC simulations. In particular, we have varied the pre-plasma scalelength in the range\n0.03-0.12 um and found that it did not significantly affect the fast electron properties and the\nfilamentary structures that develop in the dense plasma.\nWe have added a statement regarding the pre-pulse energy and laser contrast in the Methods\n(“Laser contrast and pre-plasma formation, lines 321 - 338 in revised manuscript).\nFigure 4: Tundra contrast measurement for the MEC laser [E. Cunningham et al., Appl. Phys.\nLett. 114, 221106 (2019)].\nThe Ms \"Time-resolved X-ray imaging of the current filamentation instability in solid density\nplasmas\" reports about imaging laser-matter interaction with very high time resolution using the\ncombination of a FEL and probing laser-matter interaction. The topic of radiography is always an\ninteresting topic within the community as far as it can unlock some bottlenecks within this\nresearch area. As such, the paper could be interesting for the community. However, probing of\nmatter using X-rays has already been done in the past, such as using laser-accelerated protons\nto resolve quick electromagnetic phenomena (e.g. see the PRL by Romagnani 2005). For the\nmoment it is therefore not completely clear, what superiority the quite complex setup used by\nthe author has compared to what has been used in the past and what additional information can\nbe retrieved with this complex setup in addition to what other methods would be able to detect. I\nthink that authors should emphasize this in their Ms such as to convey a clearer message and\nshow superiority of their (quite complex) setup compared to \"easier\" solutions using only X-rays\nor laser-accelerated protons.\nWe thank the Reviewer for the careful consideration and for recognizing the interest of our work.\nWe are happy to further clarify the novelty and importance of our measurements. We must\nstress that the measurements of the sub-micron density modulations in the solid-density plasma\nprovided in our work are the first of its kind and transform our ability to characterize the\nevolution of the filamentation instability in these high-energy-density conditions as we will\ndescribe in detail below. The use of an XFEL has critical advantages over optical and\nlaser-driven X-ray and proton sources for probing sub-micron modulations (associated with\nkinetic instabilities) in solid-density plasmas, including:\n1) Spatial Resolution: Kinetic plasma instabilities grow on very small spatial scales which\nare very challenging to probe at solid density. In the case of the filamentation instability,\nas we show in our work the most unstable mode has a wavelength of ~1 μm, which\nmeans that to resolve the density modulations we need a source size smaller than 0.5\nmicrons. Unfortunately, most laser-driven proton radiography and X-ray backlighters\ncurrently are still limited to source sizes of multiple micrometers [e.g. H.-S. Park et al.,\nPhysics of Plasmas 15.7, p. 072705 (2008); K. Vaughan et al., High Energy Density\nPhysics 9.3, pp. 635–641 (2013); P. K. Singh et al., Scientific Reports 12, p. 8100\n(2022)], which are not sufficient for resolving the small density filaments. Additionally,\nlaser-generated X-ray sources do not present the required spectral coherence to operate\neffectively in a microscopy setup as presented in our manuscript. The large bandwidth of\nthese sources would lead to blurring of any features of that size, meaning that even in a\nsetup aimed at compensating for the relatively large source sizes of laser-generated\nX-ray sources, the required spatial resolution could not be reached. Consequently, the\nsub-micrometer resolution (~200nm) of the MXI, which is achieved in our experiment\ndue to the brightness (large signal-to-noise), coherence and effective source size of the\nLCLS is absolutely crucial for measuring these small density fluctuations and at the\nmoment cannot be matched by laser-generated secondary sources;\n12\n2) No Deflection by Surface Fields: Previous studies [e.g., Romagnani et al., Phys. Rev.\nLett. 95, 195001 (2004); Quinn, K. et al., Phys. Rev. Lett. 108, 135001 (2012);\nAlbertazzi, B. et al., Rev. Sci. Instrum. 86, 043502 (2015); Ruyer, C. et al., Nature\nPhysics 16, 983–988 (2020)] have used proton radiography to probe the electromagnetic\n(EM) fields, associated with the filamentary instabilities in laser-plasma interactions.\nHowever, these could not resolve the filamentary fields in the solid-density plasma near\nthe interaction region, which is the most relevant to understand the development of the\ninstability and fast particle transport. The reasons for this are both the limited spatial\nresolution (as argued in the previous point) and the fact that other sources of EM fields in\nthe interaction, including surface fields, will deflect charged particles\npreventing/corrupting the measurement. For these reasons, these previous studies were\nlimited to probing the fields either in the lower-density expanding plasma or foam or\naway from the interaction region, where the spatial scale of the filaments becomes of the\norder to 50 micrometers and can be resolved by laser-driven secondary sources. This\nunderscores the critical importance of X-ray lasers, and of our results in particular, to\nunderstand not only the microphysics and nonlinear dynamics of the instability in\nsolid-density plasmas, but also how it will affect particle and energy transport in such\ndense plasma.\n3) Probing Electron Density, Not Fields: Finally, our measurements are unique in that\nthey probe directly the density and not the fields. It is due to that capability that we were\nable to demonstrate for the first time that electrostatic effects and ion motion play an\nimportant role in the development of the filamentation instability in laser-solid\ninteractions. As argued in our paper, if that was not the case, i.e. for a purely\nelectromagnetic instability, there would be no total density modulations and no\nfilamentary X-ray images would be obtained. Proton radiography is sensitive to\nelectromagnetic fields, rather than electron density, and as such the importance of\nelectrostatic and ion motion effects would not be revealed by proton probing, even if it\ncould deliver the required spatial resolution.\nFurthermore, a significant benefit of XFELs lies in reliable, high-repetition rate data acquisition,\nwhich during our experiment played a crucial role in generating the high-resolution, clean\nimages presented in the manuscript. These images were only made possible by taking\nhundreds of reference shots of the X-ray background and the static wire target for effectively\nremoving the X-ray background and revealing small density fluctuations within the target.\nFor these reasons, these are clearly first-of-a-kind measurements of density fluctuations\nassociated with the current-filamentation instability in solid-density plasmas, which at the\nmoment are only possible with an XFEL through its unique combination of coherence, flux,\nhigh-repetition rate and spatial resolution of just a few hundred nanometers that cannot be\nmatched by existing laser-driven sources.\ncomplex. In fact, following our experiment, several X-ray imaging techniques are now\nestablished as standard configurations for experiments at both LCLS and EuXFEL.\nWe have modified the manuscript to better emphasize the strong relevance of X-ray imaging for\ninvestigating solid density laser-plasma interactions (lines 74 - 86 in revised manuscript).\nAbstract: authors write \"Our findings indicate that magnetic fields on the order of 10 megagauss\nare produced, which will help confine high-energy electrons within the filaments with important\nimplications for the transport of energetic particles in plasmas.\"\nWhat about electric fields ? Are they not of importance ? What implications can this have ?\nWhat applications have authors in mind that could have an impact on this ?\nThe electrostatic component of the electric field, produced by space-charge effects, is\nshort-lived as it will serve primarily to excite the ion motion needed to maintain charge\nneutralization. There will also be an induced electric field (along the filaments) associated with\nthe growth of the magnetic field. This induced field can be estimated as E ~ Γ B λ = 0.01GV/cm,\nwhere λ is the filament wavelength, B is the magnetic field and Γ is the instability growth rate.\nThis field will last for the duration of the magnetic field amplification phase (~ 1 ps) and is too\nweak to significantly affect the energy of fast electrons. Thus, the dominant field produced in the\ndense plasma is the magnetic field as seen in our simulations.\nSuch strong magnetic fields can have important implications and applications. First of all, they\ncan affect the transport of energetic electrons, since the fields are of the order of the Alfvén limit.\nThis is important for applications such as fast ignition and proton acceleration in laser-solid\ninteractions (e.g. TNSA), since both rely on the efficient transport of fast electrons in the dense\ntarget. This fundamental understanding is also relevant for the study of charged-particle\n(cosmic-ray) transport in astrophysical plasmas, where the filamentation instability is thought to\nbe driven and play an important role but cannot be directly characterized [M. V. Medvedev, ApJ\n540 704 (2000); A. Spitkovsky, ApJ 682 L5 (2008)]. These strong magnetic fields could also\nenable novel radiation sources based on laser-solid interactions, potentially with very high\nenergy and brilliance [A. Benedetti et al., Nature Photonics 12, pages 319–323 (2018); D. J.\nStark et al., Phys. Rev. Lett. 116, 185003 2016].\nWe have added a statement regarding the implications of such strong fields in the conclusion\nparagraph (lines 276 - 283 in revised manuscript).\nPage 3: the resolution seems to be 50 fs and <200 nm. Again, what would be the resolution\nusing laser-accelerated protons (that can probe at ps time frames) or laser-accelerated X-rays,\nis a resolution of 200 nm required ?\nWe have addressed these requirements in our first response. As we explained above, indeed\nsub-micron resolution is needed, but this is not the only requirement. To directly probe these\n14\nfilamentary density structures at solid density we need a combination of penetrating radiation,\nsource size, coherence, flux and bandwidth that at present is not possible with laser-driven\nsecondary sources, underscoring the importance of the measurements provided in our work.\nPage 5: it is not completely clear what \"The temporal evolution of the filament wavelength\"\nis indicating ? How has this information been retrieved from Fig. 2 ? Maybe a small explanation\n(arrows etc) could help the reader understand the underlying calculations.\nFrom linear theory, we expect a dominant, well-defined wavelength (the fastest growing) to\ndevelop in the density filaments. By analyzing the X-ray images at different times, we observe\nthat this wavelength (the separation between filaments) is changing over time. This is an\nimportant nonlinear effect that has been theoretically associated with the merging of filaments\nvia the magnetic force but never verified experimentally. For this reason, we have analyzed how\nthe wavelength of the filaments grows over time, showing that it is consistent with the\nfilament merging model and revealing that ions play a critical role in setting the time for filament\nmerging, which had not been realized before.\nThe evolution of the filament wavelength was measured using lineouts of the X-ray images\ntaken at different pump-probe delays. Each maximum and minimum of the signal is individually\nfitted by a Gaussian function and grouped pair-wise. The distance between each pair is then\nused to extract a distribution of filament wavelengths.\nWe have improved the manuscript to introduce the methodology used to retrieve the filaments\nwavelength from the X-ray images, added an arrow in Extended Figure 3 in the revised\nmanuscript and included a reference to the Methods in the main text, where the complete\nmethodology is described (lines 167 - 168).\nPage 7: the statement \"These experimental results reveal a critical importance of electrostatic\neffects and ion motion in the dynamics of the asymmetric current filamentation instability\" - how\nis this importance revealed? The ion motion seems to be a new information, how is that\ninformation retrieved?\nWe would like to clarify that in the context of the instability growth, electrostatic effects and ion\nmotion are directly related to each other. For a purely electromagnetic filamentation instability,\nthe total plasma density modulation would be zero, i.e. the density modulation of the fast\nelectrons would be exactly compensated by the density modulation of the background electrons.\nIf that were the case, as considered in most previous theoretical studies of the instability,\nX-ray images would not show any filamentary density modulation. Thus, by measuring filamentary\ndensity modulations we are showing that electrostatic effects are indeed present.\nThese effects arise due to the density and temperature asymmetry between fast and\nbackground electrons, which cause them to pinch in the self-generated magnetic fields at\ndifferent rates. This causes a charge imbalance and therefore for the instability to grow,\nbackground ions will need to move in order to maintain the plasma quasi-neutral. This has very\n15\nimportant consequences for the growth of the instability, as it will significantly slow down its\nevolution. This is shown in Fig. 4d in the manuscript, where we can see that the growth rate is\nreduced by more than one order of magnitude by electrostatic effects, and that the resulting\ngrowth time (~ps) is consistent with the time at which the filaments are observed experimentally.\nWe further show that ion motion is important in the nonlinear (merging) dynamics of the\nfilaments. As discussed in the paper, if filament merging involved only electrons (as previously\nconsidered, e.g. in [M. V. Medvedev et al., ApJ 618 L75 (2005)]), the merging time would be ~\n85 fs (for the copper target), whereas the measured merging time is ~11 ps consistent with the\nion motion.\nWe have revised the manuscript (lines 157 - 166, 187 - 188) to better convey this important\naspect of our work.\nPage 8 (and theory): it would be interesting to compare the PIC results to the experimental\nresults. Authors state that there is a good agreement in the text, but finally there is no figure\ncomparing the results, which would strengthen their statement, even visually.\nWe thank the referee for this suggestion. Indeed, we have used PIC simulations to investigate\nthe relative contribution of electrostatic effects/ion motion and resistivity, which cannot be easily\nseparated in the experiment. In particular, by comparing the PIC simulation results, the linear\ntheory and the experimental measurements, we can contrast different models describing the\nmicrophysics of the filaments in the solid density target. In the manuscript we discuss this\ncomparison and the good agreement obtained in three important metrics:\n1. Filament wavelength. The filament wavelength measured experimentally as filaments\nare first observed is ~1 micron, as can be seen in Figures 2 and 3 in the manuscript.\nThis is consistent with both the wavelength measured in the PIC simulation (when\nincluding collisional effects; Fig. 4b) and theory (Fig. 4d), which is also ~1 micron. A\nvisual comparison can already be made between Fig. 2 and Fig. 4b. Following the\nReviewer’s suggestion, in order to make this comparison clearer, we have added the\nexperimental measurement to Figure 4d. In our opinion, the most important aspect of\nthis comparison is that, in simulation and theory, we show that collisions have a strong\nimpact in determining the filament wavelength and must be taken into account to match\nthe experimentally measured values;\n2. Filamentation growth rate. We observe experimentally that filaments arise 1.5-2 ps\nafter the laser-target interaction. This time scale is reasonably consistent with the\nmeasured growth time of the instability in the simulations, which is 1.3 ps (the inverse of\nthe measured growth rate of 0.8ps-1). While we do not measure a growth rate directly in\nthe experiment, we can compare this with the time at which filaments are first observed.\nWe have added this to Fig. 4d, where the new green triangle marker indicates the time\nfilaments are first observed in the experiments and the corresponding wavelength. We\ncan see that there is good agreement with the simulation results (red marker);\n16\n3. Magnetic field. In the experiment we estimate the magnetic field amplitude by\ncomparing the temporal evolution of the filament’s wavelength with an analytical model\nfor filament merging. From this, we obtain a value of 7-10 MG. These values are\ncomparable to the magnetic field amplitude measured in the simulations, which reach\nvalues of 7 MG as seen in Figure 4c (black curve).\nWe have modified Figure 4d in the revised manuscript to include a data point for the\nexperimentally measured wavelength and time at which the filaments appear in the X-ray\nimages, to put our measurements in perspective with the results obtained from PIC simulations\nand linear theory (Fig. 4d and associated caption in revised manuscript).\nPage 9: Author mention: \"Our results reveal the combined importance of space-charge and\nresistive effects on the growth and nonlinear dynamics of the current filamentation instability.\" -\nWhat does this mean ? Are the two, space-charge and resistive effects, individually important,\nor only its combination ? What is the impact on the filamentation individually.\nThis point is discussed between lines 232 and 274 in the revised manuscript, right before the\nsentence quoted by the Reviewer and illustrated in Fig. 4d. As we can see when comparing the\nsolid (no space-charge) and dashed black (with space-charge) curves in Fig. 4d, space-charge\neffects will primarily slow down the instability, reducing its growth rate by more than one order of\nmagnitude. Collisional effects on the other hand will primarily shift the most unstable filamentary\nmode to larger wavelengths, as can be seen by comparing black (no collisions) and red\n(collisions) circular markers in Fig. 4d. Thus, each of these processes is individually important\nand to match the experimentally observed growth time and wavelength we need to include both,\nindicating they both impact the development of the filamentation instability in solid-density\nplasmas. This is now clearer with the addition of the green marker indicating the experimental\nmeasurements in Fig. 4d, allowing for a more direct visual comparison as suggested by the\nReviewer.\nWe would like to highlight that this is an important new result that had not been established\nexperimentally before our work.\nIn general, the main message of the paper is not fully clear: Initially it seems that the main\nmessage is about a new imaging diagnostics (\"Time-resolved X-ray imaging\". Then it evolves to\nthe importance of magnetic fields in plasma and finally to filamentation issues, space-charge\nand resistive effects. I would encourage the authors to find a guideline (\"fil conducteur\") related\nto what they want to say within the Ms and build the content/results/discussion accordingly.\nWe appreciate the Reviewer’s suggestion. We note that these aspects are all connected. As\ndiscussed in our response, probing the filamentation instability in the interaction region of the\nintense laser with a solid plasma has been extremely challenging. By using a high-brightness\nand narrow-bandwidth XFEL we have managed to resolve for the first time the spatio-temporal\n17\nevolution of the plasma density filaments in the interaction. This revealed very important aspects\nfor the understanding of this instability that were not established before:\n1) Space-charge effects and ion motion are important, slowing down the instability by more\nthan one order of magnitude when compared to the purely electromagnetic case;\n2) Collisional effects are important in determining the wavelength of the filamentation\ninstability at solid-density plasmas;\n3) Strong magnetic fields, of order 10 MG, are produced in the filamentary region that can\naffect fast electron transport.\nThese results can have important implications to applications that range from studying\nrelativistic astrophysical plasma instabilities in the laboratory for the first time to energy transport\nin inertial confinement fusion plasmas to novel high-energy radiation sources.\nWe have revised the manuscript, including the abstract (lines 24 - 29 and 32-34 in revised\nmanuscript), introduction (lines 74 - 86 in revised manuscript), and conclusions (lines 276 - 283\nin revised manuscript) to make this clearer. We have also added a new paragraph at the end of\nthe introduction (lines 87 - 94 in revised manuscript, before starting discussing the experimental\nsetup) to discuss the innovative aspects of the measurements and the important findings it\nrevealed.\n18"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I have read carefully the revised manuscript \"Time-resolved X-ray imaging of the current filamentation instability in solid-density plasmas\" and the response of author to the referees' remarks. I have found my critics well addressed, as well as I've appreciated the detailed and clear responses to the rigorous remarks of the other referees. In the current form I can recommend the manuscript for publication as an article in Nature Communications.", "reviewer_2": "The authors of \"Time-resolved X-ray imaging of the current filamentation instability in solid-density plasmas\" have responded to my previous review and have added further explanations and references. However on the whole I do not believe that this paper is suitable for Nature Communications. The diagnostic is very impressive, however the Weibel instability has been explored in many other previous papers.\nIn addition there are still many aspects about this explanation that are not convincing including the density of ion perturbations probed, the duration of the filamentary structures, the length of the filaments and the 2D numerical modeling which lasts only 2 psec. The observed structures are much more likely to be due to hydrodynamics on the wire surface or due to the electrothermal instability as in previous observations.", "reviewer_3": "The authors made significant efforts for clarifying the concerns of the reviewers and the paper is now much more robust and understandable. Well done.\nFrom my side, the logical sequence is still difficult to follow: the paper being very long (10 page), a small introductory sentence guiding the reader in the \"Result\" section would help, but his is up to the authors."}, "author_response": "We thank the Reviewers for carefully assessing the revisions to our manuscript and for their\npositive feedback. Below, we reproduce the Reviewers’ comments (in black) and provide our\nresponse (in red).\nWe believe that we addressed all the Reviewers’ concerns and our manuscript is now suitable\nfor publication in Nature Communications.\nReviewer #1:\nI have read carefully the revised manuscript \"Time-resolved X-ray imaging of the current\nfilamentation instability in solid-density plasmas\" and the response of author to the\nreferees' remarks. I have found my critics well addressed, as well as I've appreciated the\ndetailed and clear responses to the rigorous remarks of the other referees. In the current\nform I can recommend the manuscript for publication as an article in Nature\nCommunications.\nWe thank Reviewer #1 for the encouraging remarks on the revised manuscript.\nReviewer #2:\nThe authors of \"Time-resolved X-ray imaging of the current filamentation instability in\nsolid-density plasmas\" have responded to my previous review and have added further\nexplanations and references. However on the whole I do not believe that this paper is\nsuitable for Nature Communications. The diagnostic is very impressive, however the Weibel\ninstability has been explored in many other previous papers.\nWe appreciate Reviewer #2 for their comments. While there have been previous studies of\nthe Weibel instability, as we discussed in the previous report, none could directly resolve the\ndevelopment of this instability in the solid density plasma region. Indeed the power of a\nX-ray Free Electron Laser presents a transformative opportunity to probe high energy\ndensity plasmas in an unprecedented way as demonstrated in our work. It is also important\nto make clear that the physics we have uncovered, including the interplay of space charge\nand collisional effects and the non-linear evolution associated with filament merging, is of\nprimary importance to understand the development and impact of the instability and had\nlong been elusive. Thus, our results provide a significant advance that in our opinion\ndeserves publication in Nature Communications.\nIn addition there are still many aspects about this explanation that are not convincing\nincluding the density of ion perturbations probed, the duration of the filamentary structures,\nthe length of the filaments and the 2D numerical modeling which lasts only 2 psec. The\nobserved structures are much more likely to be due to hydrodynamics on the wire surface\nor due to the electrothermal instability as in previous observations.\nIt would have been helpful if the Reviewer had specified which aspects of our explanation\nwere not convincing. In response to the previous report we have provided a thorough and\ndetailed explanation of these aspects, as also recognized by the other Reviewers, which in\nour opinion have addressed all the previous points of Reviewer #2. Regarding possible\nhydrodynamic structures on the wire surface we reiterate that due to the high contrast of\nthe laser there is no time for hydrodynamic processes to develop on picosecond timescales\nassociated with the filamentary structures in our experiment. This is further confirmed by\nthe clear absence of filamentary structures before the arrival of the main pulse. Finally, as\nalso discussed in our response to the previous report, if either hydrodynamic or\nelectrothermal instabilities were to develop in the low-density region outside of the target,\nas discussed in previous experiments using nanosecond lasers (which have very different\nconditions from ours), X-ray probing would not be sensitive to these. Thus, these are not\nconsistent with our observations.\nReviewer #3:\nThe authors made significant efforts for clarifying the concerns of the reviewers and the\npaper is now much more robust and understandable. Well done.\nFrom my side, the logical sequence is still difficult to follow: the paper being very long (10\npage), a small introductory sentence guiding the reader in the \"Result\" section would help,\nbut his is up to the authors.\nWe thank the Reviewer for their positive comments and their suggestion. Based on the\nprevious report we had already extended the paragraph before the “Results” section to\nguide the reader on the main findings of our work. Following the Reviewer’s suggestion we\nhave now added subsections to the “Results and Discussion” section to further help the\nreader through the structure of our manuscript."}]

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