# Ion acoustic instability resulting from suprathermal electrons generated by stimulated Raman scattering in laser-plasma interactions

C. Rousseaux, S. D. Baton, K. Glize, L. Lancia, D. Bénisti, G. Sary, L. Gremillet

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C. Rousseaux, S. D. Baton, K. Glize, L. Lancia, D. Bénisti, et al., Ion acoustic instability resulting from suprathermal electrons generated by stimulated Raman scattering in laser-plasma interactions. Physical Review Letters, 2025, 134 (16), pp.165102. 10.1103/PhysRevLett.134.165102. hal-05042296v2

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Ion acoustic instability resulting from suprathermal electrons generated by stimulated
Raman scattering in laser-plasma interactions

C. Rousseaux,¹,* S. D. Baton,² K. Glize,³ L. Lancia,² D. Bénisti,¹,⁴ G. Sary,¹,⁴ and L. Gremillet¹,⁴,†

¹CEA, DAM, DIF, F-91297 Arpajon, France

²LULI - CNRS, CEA, Sorbonne Université, Ecole Polytechnique,
Institut Polytechnique de Paris - F-91128 Palaiseau cedex, France

³Central Laser Facility, STFC Rutherford Appleton Laboratory, Harwell Oxford, OX11 0QX, United Kingdom

⁴Université Paris-Saclay, CEA, LMCE, F-91680 Bruyères-le-Châtel, France

(Dated: April 22, 2025)

Using space-time-resolved Thomson scattering, we investigate experimentally the development of electrostatic waves in a preformed undercritical helium plasma driven by a 1.059 µm laser pulse of ~1.5 ps duration and ~10¹⁵⁻¹⁶ W cm⁻² mean intensity. We observe the excitation of intense ion plasma waves (IAWs) over a broad wavenumber range, distinct from what is expected from stimulated Brillouin scattering or Langmuir decay instability. These IAWs are correlated with the occurrence of backward stimulated Raman scattering (B-SRS) at earlier times but persist much longer than the B-SRS-driven electron plasma waves. Particle-in-cell modeling reveals that they result from an ion acoustic instability triggered by the bulk electrons drifting against the ions to neutralize the current carried by the B-SRS-generated hot electrons. The properties of the simulated unstable IAWs match well with the measurements.

*Introduction.*— Suprathermal electrons are commonly produced in nonlinear laser-plasma interactions, especially due to laser parametric instabilities like stimulated Raman scattering (SRS) or two-plasmon decay [1–6]. Their generation then arises, via trapping or wave breaking, from the electron plasma waves (EPWs) excited by the instabilities. An accurate understanding of how these hot electrons could impact the subsequent laser interaction and the plasma kinetics or hydrodynamics is critical in many areas of research involving laser-produced plasmas, including inertial confinement fusion (ICF) [7, 8], laboratory astrophysics and, more broadly, high-energy-density physics [9–11].

Suprathermal electrons can affect the laser-plasma in-
teraction, and notably the very instability that spawned
them, through various kinetic mechanisms. By making
the electron distribution function (EDF) deviate from a
Maxwellian throughout the region they occupy, they usu-
ally tend to lower the Landau damping rate of the EPWs
driven by backward SRS (B-SRS) [12], thereby favor-
ing the growth of the latter instability [13, 14]. As they
travel, they can mediate the interaction between neigh-
boring speckles in smoothed laser beams. In particular,
they can alter Landau damping [15] or drive enhanced
density fluctuations [16, 17] in weak speckles that would
be SRS-stable in isolation, resulting in increased overall
SRS activity. In addition, by imparting a temperature
anisotropy to the EDF, they can induce a Weibel insta-
bility amplifying magnetic field perturbations [18–20].

A largely overlooked aspect of hot electrons generated
by parametric instabilities is their kinetic coupling to
plasma ions. In this Letter, we provide the first ex-
perimental and numerical evidence that the hot-electron
fluxes produced by B-SRS in preformed undercritical he-
lium plasmas exposed to 1.059 µm, 1.5 ps laser beams

with intensities ranging from a few 10¹⁵ W cm⁻² to a few
10¹⁶ W cm⁻² can induce a return current (carried by the
bulk plasma electrons) large enough to trigger an ion
acoustic instability (IAI) [21, 22]. The excited ion acous-
tic waves (IAWs), characterized experimentally through
space-time-resolved Thomson scattering, are found to
grow within a few 1–10 ps after the laser peak and B-SRS
saturation. Their broad wavenumber range markedly
differs from that of the IAWs driven by Langmuir de-
cay instability of B-SRS EPWs or backward stimulated
Brillouin scattering (B-SBS). Particle-in-cell (PIC) simu-
lations satisfactorily reproduce these results and further
disclose that the IAI-induced waves can generate a tail of
energetic (up to ~ 50 keV) ions in the direction opposite
to the laser propagation direction.

This process of efficient IAW generation in plasmas
driven at moderate laser intensities bears resemblance to
the so-called return-current instability, long investigated
as a possible source of energy flux limitation [23–25] and
enhanced laser absorption [26] in ICF plasmas due to
temperature gradients. However, this instability, which
has only recently been observed experimentally [27], orig-
inates from the *collisional* distortion of the EDF in re-
sponse to the current associated with the heat-carrying
electrons. This contrasts with the essentially *collisionless*
scenario reported here.

*Experimental setup.*— The experiment was performed at LULI’s ELFIE laser facility. The target plasma was created by heating a helium gas puff of ~ 1.3 mm diameter with a 40 J, long-pulse (~ 550 ps FWHM), 1.059 µm (1ω₀), smoothed laser beam. The plasma had an electron density of nₑ ≃ 0.04–0.07 n_c (n_c ≃ 10²¹ cm⁻³ is the critical density at the fundamental wavelength λ₀ = 2πc/ω₀ = 2π/k₀ = 1.059 µm) and an electron (ion) temperature close to 250 eV (70 eV) [28]. The short-pulse
<figure><img src="image_2.png" /><figcaption>FIG. 1. Space-time-resolved Thomson-scattered signals from IAWs with near-parallel wavenumbers $k_{IAW}/k_0 \simeq 2.9 \pm 0.14$, as obtained with the long-pulse $2\omega_0$ probe. The mean electron plasma density and laser intensity are $n_e \simeq 7 \times 10^{19} \text{cm}^{-3}$ and $\langle I \rangle \simeq 3.2 \times 10^{16} \text{W cm}^{-2}$ in panel (a) and $n_e \simeq 4 \times 10^{19} \text{cm}^{-3}$, and $\langle I \rangle \simeq 2.1 \times 10^{16} \text{W cm}^{-2}$ in panel (b). The origin of the laser propagation axis (x) is set to the best-focus position. The noise signal at the plasma entrance ($-1 \lesssim x \lesssim -0.5 \text{ mm}$) is caused by the $2\omega_0$ probe beam.</figcaption></figure>

($\sim 1.5$ ps FWHM) 1$\omega_0$ pump beam, focused by an $f/9$ off-axis parabola to a $\sim 15$ $\mu$m FWHM spot (in vacuum), interacted with this plasma by the end of the heating beam. The mean intensity $\langle I \rangle$ of the pump beam was varied from a few $10^{15} \text{W cm}^{-2}$ to a few $10^{16} \text{W cm}^{-2}$.

Three Thomson-scattering diagnostics were fielded to characterize the IAWs and EPWs driven in the irradiated plasma. The first used a frequency-doubled ($2\omega_0$), long-pulse ($\sim 500$ ps), low-energy portion of the heating beam to probe IAWs propagating nearly parallel to the pump beam, with wavenumbers $k_{IAW}/k_0 \simeq 2.9 \pm 0.14$ (this range corresponds to the angular aperture of the diagnostic). The scattered light, spatially resolved along the laser propagation axis, was then directed through the slit of a streak camera with 5 ps time resolution. The second diagnostic utilized a short-pulse ($\sim 0.3$ ps) 3$\omega_0$ probe to monitor both IAWs and EPWs with parallel wavenumbers $k_{IAW}/k_0 \simeq 2.1 \pm 0.8$ and $k_{EPW}/k_0 \simeq 1.9 \pm 0.7$. These ranges cover the resonant regions of B-SBS and B-SRS, respectively. The signal was collected by an imaging spectrometer, providing spatial resolution along the laser path. The EPW spectrum allowed us to infer the electron density profile seen by the pump beam. The third diagnostic was similar to the second but used a $\sim 0.4$ ps duration, $2\omega_0$ probe to measure the IAW and EPW spectra at lower wavenumbers, around $k_{IAW}/k_0 \simeq 1.5 \pm 0.7$ and $k_{EPW}/k_0 \simeq 1.3 \pm 0.5$. On some shots, the pump beam was smoothed with a random phase plate, leading to a larger ($\sim 120 \mu$m) focal spot. This increased the interaction volume and consequently the intensity of the scattered light signal.

**Experimental results.**— Figures 1(a,b) show two typical spatially and temporally resolved Thomson-scattering measurements of IAWs with near-parallel wavenumbers, $k_{IAW}/k_0 = 2.9 \pm 0.14$. The two images correspond to

different plasma densities and (monospeckle) laser intensities. The thin horizontal streaks at the time origin represent the 2nd harmonic emission of the ps pump beam propagating through the plasma, and thus indicates the onset of strong laser-plasma interaction (within the 5 ps resolution of the streak camera).

The subsequent IAW signals exhibit two remarkable features. First, they have a surprisingly high amplitude (up to $\sim 100 \times$ the detectable level, see Fig. 3) at wavenumbers unexpected from standard mechanisms like B-SBS ($k_{IAW} \simeq 2k_0$) or Langmuir decay of the B-SRS-driven EPWs ($k_{IAW} \simeq 2k_{EPW} \simeq 3.3k_0$ where $k_{EPW} \simeq 1.65k_0$ is the wavenumber of the resonant B-SRS EPWs under present conditions). Second, they develop well after the peak interaction, spreading over the region of intense laser irradiation and reaching their maximum ($\sim 0.5$ mm) longitudinal extent and amplitude after a few 10 ps. In detail, the FWHM duration (or lifetime) of the IAW signal has been found to increase approximately linearly with the laser intensity, from $\sim 5$ ps at $\langle I \rangle \simeq 2 \times 10^{15} \text{W cm}^{-2}$ to $\sim 50$ ps at $\langle I \rangle \simeq 3 \times 10^{16} \text{W cm}^{-2}$. The latter value largely exceeds the transit time of scattered electromagnetic waves across the mm-scale interaction region, so an electromagnetic origin for the IAW signal can be safely ruled out.

Figures 2(a,b) provide additional information with short-pulse ($\sim 0.4$ ps) Thomson-scattered spectra spatially resolved along the laser path. Each spectrum corresponds to a different laser shot and a different peak-to-peak time delay between the probe and pump beams. Although the absolute temporal accuracy is limited by the probe beam duration (approximately $\pm 0.2$ ps here), the relative delay between consecutive shots is highly accurate due to the use of a micrometric optical delay line and jitter-free synchronization of the beams (all originating from the same laser source).

Figure 2(a), taken 1.9 ps before the laser peak, reveals a broad spectrum of EPWs generated by B-SRS across the focal plane. This spectrum is most intense around wavelengths $\lambda \simeq 580-600$ nm but extends nearly down to the probe laser wavelength ($\lambda \simeq 529.5$ nm), a consequence of B-SRS transitioning to stimulated Compton scattering (SCS) [29] in the wave-breaking regime [30]. Notably, two faint features are also visible around the probe wavelength, indicative of IAWs with wavenumbers $k_{IAW}/k_0 \simeq 1.5 \pm 0.7$. Their appearance coincides with the large spectral broadening of the EPWs. Figure 2(b) shows that only 2 ps later (0.1 ps after the laser peak), the B-SRS EPWs have mostly decayed while the intensity of the IAW features has increased by a large amount (a factor of $\sim 12$). These data align with previous observations of rapid saturation and decay of B-SRS EPWs during similar ps laser interactions [28, 31]. Moreover, they suggest that the anomalously strong IAWs uncovered in Fig. 1 have a broad wavenumber range, extending from below to above $2k_0$, and support their delayed devel-
<figure><img src="image_3.png" /><figcaption>FIG. 2. (a,b) Thomson-scattered spectra from B-SRS-induced EPWs ($k_{EPW}/k_0 \simeq 1.3 \pm 0.5$) and IAWs ($k_{IAW}/k_0 \simeq 1.5 \pm 0.7$), spatially resolved along the laser propagation axis (x) at two times: (a) 1.9 ps before the peak of the pump beam and (b) 2 ps later. The same square-root color scale is used to attenuate the contrast of the images, showing that IAWs start to grow precisely when EPWs saturate due to wave breaking. (c) Thomson-scattered spectra from IAWs resolved in scattering angle ($\phi$, left axis) or equivalently in wavenumber ($k$, right axis), taken 8.5 ps after the laser peak. All these spectra were obtained with the short-pulse ($\sim 0.4$ ps) $2\omega_0$ probe for $n_e \simeq 4 \times 10^{19}$ cm$^{-3}$ and $\langle I \rangle \simeq 8 \times 10^{16}$ W cm$^{-2}$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 3. Intensity of the Thomson-scattered signals from IAWs with $k_{IAW}/k_0 \simeq 2.9 \pm 0.14$ (blue circles, left axis) and B-SRS-driven EPWs with $k_{EPW}/k_0 \simeq 1.9 \pm 0.7$ (red squares, right axis with same dynamic range as in left axis) as a function of the mean laser intensity $\langle I \rangle$. Measurements are compared for (a) a monospeckle pump beam ($\sim 15$ µm FWHM) and (b) a RPP-smoothed pump beam ($\sim 120$ µm FWHM). The electron density is $n_e \simeq 7 \times 10^{19}$ cm$^{-3}$ for all data points.</figcaption></figure>

opment relative to the main laser interaction and B-SRS activity.

A closeup of the IAW spectrum resolved in scattering angle and hence in wavenumber [in the same range as panels (a) and (b)], measured $\sim 8.5$ ps after the laser peak, is shown in Fig. 2(c). The blue-shifted feature, associated with IAWs propagating in the opposite direction to the laser beam, appears to be about twice brighter than the red-shifted feature, related to IAWs comoving with the laser beam. This asymmetry points to a preferential excitation of backward-propagating IAWs at later times. As will be demonstrated below, this results from an ion acoustic instability triggered by the bulk plasma electrons drifting backwards against the ions.

Figure 3(a) plots the intensity of the B-SRS EPWs around $k_{EPW} \simeq 1.9 \pm 0.7$ and of the IAWs around $k_{IAW} \simeq 2.9 \pm 0.14$ as a function of the mean intensity of the monospeckle laser. The EPW intensity is obtained by spatially integrating the EPW spectrum recorded by the short-pulse $3\omega_0$ Thomson probe at the laser peak.

The IAW intensity corresponds to the maximum of the space-time-resolved IAW signal from the long-pulse $2\omega_0$ Thomson probe (see Fig. 1). Both EPW and IAW quantities are measured on a single-shot basis. All data points correspond to an electron density of $\sim 7 \times 10^{19}$ cm$^{-3}$. The plot confirms the correlation between the onset of B-SRS (identified by EPW growth) and the development of non-standard strong IAWs, both occurring above about the same laser intensity of $\sim 2 \times 10^{15}$ W cm$^{-2}$. This threshold value for B-SRS is consistent with previous measurements under similar conditions [16, 28]. Figure 3(b) shows that this correlation extends to multispeckle laser beams. In this case, weaker speckles can be destabilized by stronger neighbors [16, 17, 32], which lowers the B-SRS threshold mean intensity to about $3 \times 10^{14}$ W cm$^{-2}$. The consistent appearance of those IAWs above the same threshold intensity as B-SRS further argues against their origin from B-SBS. Previous work has indeed found that for $\sim 1$ ps pump beams, the threshold for B-SBS significantly exceeds that for B-SRS [28].

*Numerical modeling.*— Our experimental findings are elucidated by a two-dimensional, fully kinetic simulation performed with the PIC CALDER code [16, 17, 33], under conditions approaching the experimental ones. The laser pulse, of wavelength $\lambda_0 \equiv 2\pi c/\omega_0 = 1$ µm, has a Gaussian intensity envelope of $2600 \omega_0^{-1}$ (1.38 ps) FWHM duration, $100 c/\omega_0$ (15.9 µm) FWHM focal spot and $4 \times 10^{16}$ W cm$^{-2}$ peak value. It propagates along $x > 0$ through a plasma composed of electrons and fully ionized He$^{2+}$ ions, with respective initial temperatures of $T_e = 250$ eV and $T_i = 70$ eV. The peak laser intensity is reached in vacuum at position $x = 3000 c/\omega_0$ (477 µm) and time $t = 5600 \omega_0^{-1}$ (2.97 ps). The plasma has a parabolic density profile extending over $5800 c/\omega_0$ ($\sim 923$ µm), with a maximum electron density of $0.07 n_e$ located at $x = 3600 c/\omega_0$ (573 µm). Further numerical details are provided in Appendix A.

In line with previous studies [16, 17, 28, 31, 34–36], the simulation predicts intense B-SRS activity during the rising edge of the laser pulse. The $\omega-k_x$ power spec-
<figure><img src="image_5.png" /><figcaption>FIG. 4. (a) Spatiotemporal power spectrum of the longitudinal electric field $E_x$, computed from the PIC simulation over $3.02 < t < 7.05$ ps and $318 < x < 477$ µm along the laser symmetry axis ($y = 0$). The solid and dashed magenta curves represent the linear IAW eigenmodes computed from the plasma distribution at $t = 3.02$ ps. (b) Time evolution of the $E_x$ fluctuation intensity integrated in spectral regions (detailed in text) associated with B-SRS EPWs (blue circles), $v_\phi < 0$ IAWs (yellow upward triangles) and $v_\phi > 0$ IAWs (red downward triangles). Solid and dotted curves correspond to on-axis ($y = 0$) and off-axis ($y = 15.9$ µm) quantities, respectively. The vertical dashed line indicates the laser peak time.</figcaption></figure>

trum of EPWs is initially concentrated around the resonant region of B-SRS ($k_x/k_0, \omega/\omega_0) \simeq (1.66, 0.25)$. However, electron trapping and subsequent wave breaking due to unstable sidebands [37, 38], occurring as early as $t \simeq t_{max} - 0.7$ ps ($t_{max}$ is the time of local peak intensity), generate a very broad turbulent spectrum extending over $0 \lesssim k_x/k_0 \lesssim 2$ and down to vanishing (positive and negative) frequencies. In parallel, the spectrum develops an elongated feature evidencing the evolution of B-SRS into SCS [30] [see Fig. 6(a) in Appendix B]. This behavior is consistent with the EPW spectrum shown in Fig. 2(a).

Figure 4(a) zooms in on the low-frequency (LF) part of the power spectrum of the longitudinal electric field, $|E_x(k_x, \omega)|^2$, recorded along the laser propagation axis ($y = 0$) over the spatiotemporal range $318 < x < 477$ µm and $3.02 < t < 7.05$ ps (i.e., $0.32 \lesssim t - t_{max} \lesssim 4.35$ ps with $t_{max} \simeq 2.70$ ps). The local mean electron density is of $\sim 0.065n_c$. The spectrum consists of two distinct IAW branches with positive and negative wave phase velocities ($v_\phi = \omega/k_x$), originating from ($k_x, \omega$) = (0, 0) and extending up to $k_x/k_0 \simeq 6$. Interestingly, the $v_\phi < 0$ branch is about ten times more intense than the $v_\phi > 0$

branch, peaking in the range $0.5 \lesssim k_x/k_0 \lesssim 3$. These results qualitatively agree with the asymmetric IAW features seen in Fig. 2(c).

To capture the dynamics of the LF plasma fields, we compute $|E_x(k_x, \omega)|^2$ over a shorter ($\sim 1$ ps) time window, and integrate it over two LF spectral regions defined by $0.8 < k_x/k_0 < 2.2$ and $0 < \omega/\omega_0 < 0.015$ or $-0.015 < \omega/\omega_0 < 0$. This is done both along $y = 0$ and $y = 15.9$ µm. For comparison, we also extract the EPW signal by integrating the power spectrum over $1.5 < k_x/k_0 < 2$ and $0.15 < \omega/\omega_0 < 0.35$. Figure 4(b) shows the time evolution of these quantities, revealing three stages in the evolution of the LF fields.

During the initial phase ($1 \lesssim t \lesssim 2$ ps), before the laser peak ($t_{max} \simeq 2.70$ ps here), the $v_\phi < 0$ (yellow upward triangles) and $v_\phi > 0$ (red downward triangles) LF signals exhibit the same strong growth ($\sim 1000 \times$ the initial level). This increase coincides with an even larger ($> 10^4 \times$) growth in the EPW signal due to B-SRS, which locally saturates at $t \simeq 2$ ps. The initial growth of the LF signals is due to the considerable spectral broadening of the $E_x$ fluctuations caused by electron trapping in the B-SRS EPWs followed by wave breaking. This turbulent spectrum, which extends down to IAW-range frequencies, consists only of electron fluctuations. This was verified by running a simulation using stationary ions, which predicted the same early rise in the integrated LF signals (see Appendix B).

The $x-p_x$ electron phase space averaged around the axis ($|y| < 15.9$ µm) at the end of this phase is detailed in Fig. 5(a). Strong trapping sets in around $x \simeq 450$ µm, giving rise to periodic, well-defined electron vortices. However, due to the rapid growth of sidebands, these vortices get disrupted and merge together [37–39]. As the electron orbits fill the inside of the vortices, this results in the formation of a rather homogeneous electron tail extending over $0.1 \lesssim p_x/m_ec \lesssim 0.4$ for $x \lesssim 390$ µm. In turn, the remaining bulk of the EDF shifts towards $v_x < 0$ to maintain current neutrality.

The second stage [see Fig. 4(b)], occurring for $2 \lesssim t \lesssim 4.5$ ps, is characterized by a pronounced (by $\sim 1000 \times$) decay of the EPW intensity due to wave breaking. Yet the most striking observation is the contrasting behavior of the $v_\phi > 0$ and $v_\phi < 0$ LF signals: while the former saturates and starts to wane, the latter continues to grow by an additional order of magnitude, eventually accounting for most of the total LF intensity. This originates from the negative drift velocity of the bulk electrons [see Fig. 5(b) for the spatially averaged EDF recorded at $t = 2.42$ ps] which has become large enough relative to the sound speed to destabilize the $v_\phi < 0$ IAWs within the probed $k$-range [22].

We confirm the onset of an IAI by numerically solving [21] the kinetic dispersion relation of the plasma modes, fitting the on-axis EDF to a sum of nonrelativistic, drifting Maxwellians and considering the unper-
<figure><img src="image_6.png" /><figcaption>FIG. 5. (a) Electron $x-p_x$ phase space at $t = 2.01$ ps, transversely averaged around the axis ($|y| < 15.9$ µm) and plotted over $318 < x < 477$ µm. (b) Electron $p_x$-distribution averaged over $318 < x < 398$ µm and $|y| < 15.9$ µm at $t = 2.42$ ps (solid blue) and multi-Maxwellian fit (dashed red). (c) Imaginary frequency vs. wavenumber for the linear IAW eigenmodes with $v_\phi < 0$ (solid blue) and $v_\phi > 0$ (dashed red), computed from the model electron distribution plotted in (b) and a Maxwellian ($T_i = 70$ eV) ion distribution. (d) Ion $x-p_x$ phase space at $t = 10.07$ ps, averaged around the axis ($|y| < 15.9$ µm).</figcaption></figure>

turbed Maxwellian ion population ($T_i = 70$ eV). The real and imaginary frequencies of the dominant IAW modes are plotted in Fig. 4(a) and Fig. 5(c), respectively. These modes correspond to the continuation of the standard (least damped) IAWs that would exist in the unperturbed plasma. While other IAW-type modes are present [21], they are all stable and exhibit a higher damping rate at $k_x/k_0 \gtrsim 3$. As expected, the IAW branch with $v_\phi < 0$ proves unstable for $k_x/k_0 \lesssim 10$, with a maximum growth rate $\omega_i \simeq 7 \times 10^{-4} \omega_0$ ($\simeq 1.3$ ps$^{-1}$) around $k_x/k_0 \simeq 2-3$. This growth rate greatly exceeds that ($\simeq 5$ ns$^{-1}$) recently reported for the collisional return-current instability using $\sim 10^{15}$ W cm$^2$ ultraviolet laser beams [27]. These predictions reasonably agree with the simulation data of Fig. 4 and the experimental observation of a $\sim 12 \times$ increased IAW intensity within $\sim 2$ ps [Figs. 2(a,b)]. The somewhat lower dominant wavenumbers seen in Fig. 4(a) likely result from the nonlinear evolution of the unstable plasma over the time frame of the Fourier transform.

Figure 5(c) also reveals that the damping rate of the $v_\phi > 0$ IAWs is locally maximized around $k_x/k_0 \simeq 3$, reaching $\omega_i \sim -5 \times 10^{-4} \omega_0$ over the range $1 \lesssim k_x/k_0 \lesssim 5$ (including the B-SBS resonant region at $k_x/k_0 \simeq 2$). This significant damping is due to the negative drift of the bulk electrons; neglecting it in our calculation would reduce the damping more than tenfold.

At the edge of the laser beam ($y = 15.9$ µm), B-SRS is weaker and so is the early increase in the two IAW signals [Fig. 4(b)]. Later on ($t \gtrsim t_{max}$), the $v_\phi < 0$ IAWs show a steady growth at a rate ($\simeq 0.3$ ps$^{-1}$) of the same order as the above estimate. The concomitant growth in the $v_\phi > 0$ IAWs may stem from induced scattering of the $v_\phi < 0$ IAWs on ions [40].

In the third stage ($t \gtrsim 4.5$ ps), both on-axis IAW signals gradually weaken due to energy transfer to plasma particles [41]. Remarkably, the ions trapped in the IAI-driven waves can be accelerated to negative velocities as high as $v_x/c \simeq -5 \times 10^{-3}$, corresponding to a kinetic energy of $\sim 50$ keV. This significant backward ion acceleration is manifest in Fig. 5(d), which plots the ion $x-p_x$ phase space at a late stage the simulation ($t = 10.07$ ps).

**Conclusions.** — Through ultrafast Thomson-scattering

measurements and PIC modeling, we have shown that the hot electrons generated by B-SRS of $\sim 10^{15-16}$ W cm$^{-2}$ ps laser pulses in undercritical plasmas consistently trigger an IAI via the induced return current. This instability amplifies a broad spectrum of IAWs within a few 1–10 ps, which are predicted to heat the negative-velocity tail of the ion distribution to tens of keV energies. Actually, IAWs are enhanced not only in the $v_\phi < 0$ region, where the IAI operates directly, but also, albeit less strongly, in the $v_\phi > 0$ region relevant to B-SBS. Our preliminary 1D simulations indicate that the IAI occurs as well under conditions close to those found in ICF experiments. In this context, the enhanced $v_\phi > 0$ IAWs could act as strong seeds for B-SBS in subsequent laser speckles [42, 43]. Still, the overall effect on B-SBS is not obvious since our analysis also predicts increased damping of the B-SBS-resonant IAWs due to the electron return current. Further investigation is therefore needed to fully understand the implications of our results for ICF, particularly in the context of $2\omega_0$ pump lasers [44] or shock ignition [45].

The authors acknowledge the LULI technical staff for their support during these experiments. L. G. thanks B. Canaud and A. Debayle for useful discussions. The simulations were performed using HPC resources at TGCC/CCRT. This work, carried out within the framework of the EUROfusion Consortium through the ToIFE project, has received funding from the Euratom research and training programme 2014-2018 under grant agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work has also been supported by the Agence Nationale de la Recherche (ANR) (UnRIP project, Grant No. ANR-20-CE30-0030).

**Appendix A: PIC simulation setup.** — The numerical domain has dimensions $L_x \times L_y = 6000 \times 2000$ ($c/\omega_0)^2$ (955 × 318 µm$^2$) with mesh size $\Delta x = \Delta y = 0.2$ c/ω₀ (0.032 µm). The time step is $\Delta t = 0.95\Delta x$. The plasma is modeled by 50 macroparticles per cell and per species, yielding a total of $\sim 3 \times 10^{10}$ macroparticles. The initial electron density profile is uniform in the transverse ($y$) direction and parabolic in the longitudinal ($x$) di-
rection, peaking at $x = 3600 c/\omega_0$ (573 µm). In detail, it ramps up from $n_e \simeq 0.023 n_c$ to $0.07 n_c$ over $100 \le x \le 3600 c/\omega_0$ and decreases down to $n_e \simeq 0.05 n_c$ at $x = 5900 c/\omega_0$. Coulomb collisions between all plasma species are treated, but an additional simulation performed without them found that their effect is negligible within the spatiotemporal scales of interest. Another simulation was run with immobile ions to assess their contribution to the early and late phases of the plasma dynamics. Particles exiting the domain are reinjected with random thermal velocities while open boundary conditions are applied to the electromagnetic fields. The simulation is run over $\sim 1.2 \times 10^5 \Delta t$ ($\sim 12$ ps).

**Appendix B: Simulation results with and without mobile ions.**— Figure 6(a) displays the power spectrum of the on-axis $E_x$ field from the reference PIC simulation over the range $2.01 < t < 2.52$ ps ($-0.689 < t_{max} < -0.185$ ps) and $318 < x < 477$ µm. The B-SRS EPWs (initially excited around $\omega/\omega_0 \simeq 0.25$ and $k_x/k_0 \simeq 1.66$) have already broken, leading to an oblique (in $\omega-k_x$ space) continuum of SCS-driven EPWs [30]. Moreover, the unstable sidebands of the nonlinear B-SRS EPWs [37–39] and the electron acoustic-type waves (EAWs) excited in the highly non-Maxwellian EDF [34, 36, 46] have given rise to a broad distribution of fluctuations, extending down to $k_x \simeq 0$ and slightly negative $\omega$. The thin line around $\omega = 0$ corresponds to IAWs. Also visible around $\omega/\omega_0 \simeq \pm k_x/k_0 \simeq \pm 0.26$ are the forward-propagating EPWs due to forward SRS (F-SRS) and the backward-propagating EPWs due to F-SRS of B-SRS light.

Figure 6(b) shows that the same spectrum is obtained when considering immobile ions, except of course for the absence of IAWs. The corresponding time evolution of the B-SRS EPW and LF signals is plotted in Fig. 6(c). Compared to the case with mobile ions [Fig. 4(b)], the B-SRS behavior is largely unchanged while the LF signals show the same initial rapid rise before decaying past the laser peak.

<footnote>* christophe.rousseaux@cea.fr</footnote>

<footnote>† laurent.gremillet@cea.fr</footnote>

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<figure><img src="image_7.png" /><figcaption>FIG. 6. (a,b) Spatiotemporal power spectra of the $E_x$ field over $2.01 < t < 2.52$ ps and $318 < x < 477$ µm along the laser symmetry axis ($y = 0$), with (a) or without (b) mobile ions. (c) Same as Fig. 4(b) but with immobile ions.</figcaption></figure>

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