The coupling of an intense laser pulse (I ≳ 10¹⁴ W/cm²) with a plasma corona is strongly affected by the onset of laser–plasma instabilities (LPIs). In both indirect-drive and direct-drive schemes of inertial confinement fusion (ICF), this can influence the efficiency and symmetry of fuel compression, and ultimately the fusion gain. Among these instabilities, backward stimulated Raman scattering (BSRS), where laser light is scattered by an electron plasma wave (EPW) in the backward direction, has been thoroughly investigated over the last 40 years. Conversely, however, side stimulated Raman scattering (SSRS) has been hardly investigated and remains poorly understood. In SSRS, laser light is scattered by an EPW in a direction perpendicular to the electron density gradient and is successively refracted to lower densities, finally exiting the plasma at large angles [Fig. 1(d)], as described below.

Early theoretical work on SSRS, dating back to the 1970s,1–3 immediately revealed the difficulty in describing a process that can have both an absolute character, with eigenmodes temporally growing at a single point, and a convective character, with wave packets growing in a resonance region along the scattered light trajectory. In addition, convective growth occurs near the turning point of scattered light where the Wentzel–Kramers–Brillouin (WKB) approximation fails,4 making theoretical analysis less straightforward.3 The first coherent picture of SSRS, derived for infinite transverse extension and negligible Landau damping of the EPW, was proposed by Mostrom and Kaufman,3 showing that wave packets rapidly grow and saturate owing to refraction out of the resonance zone, while eigenmodes grow more slowly and prevail at later times. In this model, both the resonance length L_res of the wave packets, and thus their gain, and the eigenmodes threshold scale with the electron density scale length L_n ≡ n/(dn/dz).

Although early work predicted a massive and threatening impact on ICF performance, only a few experiments in the 1980s5–7 succeeded in detecting a weak SSRS signal, which led to a rapid halt to its investigation. Possible reasons for these detection failures could be the finite size of the laser beam, limiting the growth of the instability, the high collisional absorption of scattered light before leaving the plasma, and the difficulty in measuring light scattered over a wide solid angle. Recent observations of strong SSRS at the NIF,8–10 OMEGA-EP11 and SG-II UP,12–15 driven at laser intensity in the range 10¹⁴–10¹⁵ W/cm², stimulated, however, renewed effort aimed at its experimental characterization. At the same time, recent theoretical work has explored cases of strong Landau damping,10 closer to the expected ICF conditions, and has quantified the corrections due to the limited size of the laser beam.3,16 The k-space theory, finally, has provided a more complete and coherent framework of SSRS,17,18 reproducing the extreme cases of cold and hot plasmas found in previous papers.

One difficulty in SSRS characterization, and even more in the measurement of its reflectivity, is due to its intrinsic 3D geometry, since the intensity of the scattered light is expected to depend on both the polar angle, i.e., the angle with respect to the density gradient, because of light refraction into the plasma, and on the azimuthal angle, i.e., the angle with respect to the plane of polarization. In the following, the directions in the focal plane that are parallel and perpendicular to the laser polarization will be named the p-direction and s-direction, respectively. Correspondingly, the P-plane and the S-plane will respectively indicate the plane of polarization and the plane perpendicular to it that includes the laser axis.

Following a clear observation of SSRS in a previous experiment at the Prague Asterix Laser System (PALS),19 we present here the first angularly resolved measurements of SSRS driven at laser intensities ∼10¹⁶ W/cm² that are relevant for the shock ignition scheme for ICF. The measurements, obtained with the specifically designed Octopus diagnostic, clearly show the dependence of SSRS on the laser spot extent in the s-direction. We also demonstrate that, differently from BSRS, SSRS is here unaffected by the density scale length of the plasma.

The experiment was carried out at PALS by focusing a single laser pulse (λ₀ = 438 nm, τ ≈ 300 ps) at normal incidence on plane targets by means of an f/2 lens. To investigate the effects of focal spot size and orientation with respect to laser polarization, two different random phase plates (RPPs) were used, providing either a Gaussian circular spot (RPP1, FWHM ≈ 60 μm) or an elliptical super-Gaussian spot (RPP2, FWHM1 ≈ 40 μm, FWHM2 ≈ 100 μm), resulting in comparable peak laser intensities in the range (1.0–1.4) × 10¹⁶ W/cm². Since SSRS is expected to grow more strongly along the s-direction,2,16 during the experiment RPP2 was rotated to align the longer axis along (configuration RPP2-P) and across (RPP2-S) the laser polarization [Figs. 1(a)–1(c)].

The dependence of LPI on density scale length was investigated by varying the target thickness, going from “thick” multilayer targets—with 20, 60, and 100 μm Parylene-C ablators, followed by a 7 μm Cu tracer layer—down to “thin” Parylene-N foils of 2.16, 2.99, and 5.41 μm (hereinafter referred to as 2, 3 and 5 μm foil targets). Target thickness was measured by spectral reflectance and contact probe techniques with an uncertainty <10 nm. A ∼50 nm Al coating was deposited on the laser side of both the Parylene-C and Parylene-N ablators to avoid laser propagation at early times of interaction. It is worth noting that for the “thick” targets, new material is ablated at any time during laser irradiation, and so the plasma density profile does not depend on the target thickness; accordingly, no substantial differences were observed in the features of LPI for the various thick targets in this experiment. In the following, we therefore gather them all as “thick target” shots. By contrast, for the so-called “thin” targets, the laser is expected to burn through the foil at a certain time, depending on the initial thickness, when the peak electron density falls below the critical density n_c of the laser light; from that time, no more material is ablated and the hydrodynamic evolution is determined by plasma expansion alone.20

The expected plasma conditions during laser interaction were investigated by means of hydrodynamic simulations with the 2D code CHIC,21 for the circular laser spot [Fig. 2(a)], and with the 3D code Troll22 for comparing the differences between circular and elliptical spots [Fig. 2(b)]; the discrepancies in the temperatures and density scale length values obtained with the two geometries were below 5%. Simulations show that L_n increases when the target thickness decreases from thick targets down to 2 μm [Fig. 2(a)]. At a time ≈150 ps after the laser peak and in the density region n = 0.12n_c, for which SRS instability is stronger, the calculated values of L_n are 63, 76, 86, and 115 μm for thick targets and 5, 3, and 2 μm foils, respectively. For thin foil targets, the laser burns through before the end of the laser pulse [Fig. 2(a)], making L_n locally diverge at longer times and higher densities.

BSRS was characterized by full-aperture beam (FAB) calorimetry, using a Gentec pyroelectric detector, and FAB time-resolved spectroscopy, using a Czerny–Turner ISOPLANE-160 spectrograph (Princeton Instruments) with 150 or 300 lines/mm gratings coupled to a Hamamatsu C7700 optical streak camera. SSRS was measured by the angularly resolved Octopus spectrometer [Fig. 1(e)], consisting of two circular aluminum arms, each holding 12 optical fibers placed at polar angles going from 23° to 90° to the laser beam axis. The two arms, aligned in planes at 10° (hereinafter “P”) and at 80° (hereinafter “S”) to the laser polarization, allowed simultaneous measurement of Raman spectra at different polar angles in planes near-parallel (P-plane) and near-perpendicular (S-plane) to the laser polarization. The 24 fibers were bundled to the entrance slit of a MS257 Oriel 1/4-m imaging spectrometer coupled to a 14-bit CCD, resolving SSRS spectra in the range 600–910 nm. A white light source of known emissivity was placed at the target chamber center (TCC), allowing absolute energy calibration for the signal measured by each fiber; the calibration procedure was the same as described in Ref. 13.

A typical angularly resolved spectrum obtained using the RPP1 plate and a thick target is shown in Fig. 3; the corresponding values of SRS scattered energy at each angle, integrated in the spectral range 600–800 nm and measured in joules per solid angle (J/sr), is reported in Fig. 4(c) for the P- and S-planes.

The SSRS signal can be clearly recognized in the diagonal traces visible in both planes. Here, the SRS spectra measured at each angle are different, showing a peak signal moving to shorter wavelengths for larger polar angles. This trend is a signature of SSRS, where a definite relation between the wavelength λ_SSRS of the scattered light and its exit angle θ_out holds. This light is scattered in the region 0.07n_c–0.18n_c as obtained assuming a ≈2 keV plasma temperature given by hydrodynamic simulations [Fig. 2(a)]. The experimental λ_SSRS–θ_out relation (blue dashed line), however, departs from the analytical formula θout=arctanω02/ωp2−2ω0/ωp (black dots), where ω₀ and ω_p are the laser light and plasma frequencies, which is obtained by considering light refraction in a 1D plasma profile;16 it is likely that the trajectory of SSRS light into the plasma is here affected by the curvature of the isodensity lines into the plasma. We finally note that the signal in the S-plane is much stronger–by about two orders of magnitude [Fig. 4(c)]–than that in the P-plane, as expected from theory.2

In addition to SSRS, Fig. 3 shows a signal at near-backscattering angles, with comparable intensities in the P- and S-planes. This spectrum (e.g., that observed at θ_out = 23°) is consistent with that measured by the FAB spectrometer (θ_out < 14°), showing a BSRS signal peaked at λ ≈ 700 nm, and the ω₀/2 doublet, which is a signature of two-plasmon decay (TPD) instability [Fig. 5(a)]. This shows that BSRS extends in the near-backscattering cone (n-BSRS) and progressively fades for larger polar angles. Figure 3 also shows that the ω₀/2 feature disappears rapidly with increasing polar angle, owing to the stronger refraction of light generated in the n_c/4 region. The streaked spectra in the FAB line [Fig. 5(b)] show that both BSRS and TPD are driven during the trailing part of the laser pulse at times ∼100–150 and ∼200 ps, respectively, after its peak. A similar result was obtained and modeled for BSRS in a previous experiment,23,24 showing that the strongest growth is obtained for larger values of L_n, i.e., at late times, and while the laser intensity is still significant; the late onset of TPD is therefore unexpected. Calorimetric measurements in the FAB cone obtained for circular spots on thick targets yielded a BSRS reflectivity of (0.020 ± 0.005)%, similar to the levels measured in previous work under similar conditions.23,24 We note that L_n is here an order of magnitude smaller than that expected in an ICF plasma corona, which explains the low reflectivity measured in the experiment. The total reflectivity of SSRS is not measured here.

The dependence of SSRS on focal spot shape is revealed in Fig. 6, where angularly resolved spectra measured from thick targets with RPP2-P and RPP2-S configurations are shown; the corresponding spectrally integrated SRS scattered energies (J/sr) in the P- and S-planes are reported in Figs. 4(a) and 4(e).

Although the peak laser intensity I ≈ 1.2 × 10¹⁶ W/cm² is comparable to that of Fig. 3, clear differences emerge from comparison of the spectra. First, the SSRS spectra observed for the RPP2-P shots are limited to wavelengths >700 nm, corresponding to densities higher than 0.13n_c. Second, the deviation of the experimental λ_SSRS–θ_out relation from that expected in a 1D plasma geometry is more accentuated, producing a mixture of SSRS and n-BSRS signals at small polar angles; in Fig. 6(a), for example, a strong SSRS peak at λ ≈ 750 nm is observed at θ = 23°, as plotted in Fig. 7(a). FABS spectra measured in RPP2-P shots consistently exhibited a peak at shorter wavelengths, produced by BSRS, and an additional signal at longer wavelengths, corresponding to the SSRS peak observed at near-backscattering angles. The identification of the origin of the second peak in the streaked spectra allowed us to determine the time of SSRS emission, which is ∼100–200 ps after the laser peak; the SSRS growth is therefore concomitant with those of BSRS and TPD. The strong refraction of SSRS light toward near-backscattering angles could be produced by a significant curvature of isodensity lines into the plasma as shown in Fig. 2(c); the observation of scattered light at lower polar angles could be produced, for example, by the onset of SSRS at the borders of the laser beam, where the light incidence is oblique with respect to the isodensity surface, and the side light is scattered in the backward SSRS configuration.25 In the case of shots using the RPP2-S configuration, the maximum intensity of SSRS is comparable to that obtained with RPP1; here, however, the spectrum observed at each polar angle extends to shorter wavelengths, as can clearly be seen on comparing the curves of scattered energy in the S-planes in Figs. 4(c) and 4(e). These circumstances make a clear distinction from n-BSRS more difficult. This result could be explained by refractive effects or by the excitation of SRS in a broader range of angles with respect to the density gradient, going from 90° (SSRS) down to n-BSRS. Detailed modeling of SSRS growth into realistic hydrodynamic plasma maps by using suitable ray-tracing codes11 is needed to reproduce the observed results.

A further difference revealed by comparison of Figs. 3 and 6 is that the intensity of SSRS is significantly lower in the RPP2-P configuration than in the RPP1 case, while it is slightly higher in the RPP2-S shots. For thick targets, the value obtained by adding up the spectrally integrated SSRS signals in all the fibers is in fact a factor ∼10 lower for RPP2-P shots and ∼2 higher for RPP2-S shots. Since the density scale length, calculated on the central axis, is comparable in all cases, this suggests that SSRS growth is here restricted by the size of the spot in the S-direction.

This hypothesis can be corroborated by comparing the resonance length L_res of SSRS with the size of the focal spot D in the three irradiation configurations. In this experiment, the Landau damping of the EPW is negligible for n > 0.1n_c (corresponding to λ_SSRS > 640 nm), implying that the convective growth of SSRS is limited by the refraction of scattered light out of the resonance zone, where the matching conditions of energy and momentum are fulfilled. Following the model of Mostrom and Kaufmann,3L_res can be estimated as the distance traveled by the wave packet at velocity v_g/2, where v_g is the group velocity of the scattered light, during the time of saturation t_s of the wave packet growth [Eq. (2c) in Ref. 3]; for a density of 0.14n_c (corresponding to λ_SSRS = 700 nm), we then obtain a resonance region of thickness Δz ≈ 6.7 μm and length L_res ≈ 120 μm, traveled in a time t_s ≈ 700 fs. These values are derived for a 1D plasma, depend linearly on L_n, and are calculated for thick target shots at a time 150 ps after the laser peak. For a finite focal spot size, however, L_res can be reduced by the plasma curvature if ρ_SSRS ≳ ρ_2D, where ρSSRS=2Ln[1−(ωp/ωs)2]/(ωp/ωs)2 is the radius of curvature of the scattered light trajectory (of frequency ω_s) in a 1D plasma, and ρ_2D is the radius of curvature of isodensity lines in the plasma. In that case, the scattered light moves out of the resonance region at earlier times and the growth stops at a reduced resonance length Lres′. In the case of a 1D plasma, the SSRS wave-packet saturation is produced by the density change corresponding to a longitudinal movement equal to the resonance thickness Δz. The value of Lres′ in a curved plasma can be estimated by calculating the transverse size needed by a scattered light ray, refracting in a spherical plasma with the same density scale length value, to undergo the same density decrement. Using this procedure, we obtain Lres′≈Lres(1+ρSSRS/ρ2D)−1/2.

Hydrodynamic simulations show that the radius of curvature ρ_2D of the isodensity lines depends on the time and the electron density considered. For n = 0.14n_c (corresponding to λ_SSRS = 700 nm) at a time 150 ps after the laser peak, we obtain ρ_2D ≈ 200 μm and ρ_2D ≈ 110 μm for RPP1 and RPP2, respectively. It is interesting to note that the profile along the two axes in the RPP2 plasma is similar, despite the elliptical spot, as shown in Fig. 2(b) for n_e = 0.10n_c; this effect results from the expansion of the plasma making the profile in the transverse plane quite isotropic at later times of interaction. The above ρ_2D values correspond to resonance lengths of Lres′≈80μm and Lres′≈70μm for the RPP1 and RPP2 cases, respectively. In addition to plasma curvature, the growth of the wave packet is obviously confined by the finite size D of the beam in the S-direction. We can therefore conclude that the gain ΓSSRS∝min(D,Lres′,Lres), where the dependence on L_n is lost if the beam size or the plasma curvature prevails.3 Assuming D ≈ FWHM of the laser spot, calculated along the S-direction, we note that for the RPP2-P and RPP1 shots, Γ_SSRS ∝ D; i.e., the resonance regions, and therefore the gain, are limited by the size of the focal spot (≈40 and 60 μm, respectively). By contrast, in the RPP2-S shots, where D>Lres′, SSRS growth is limited by the curvature of the plasma (Lres′≈70μm).

This picture satisfactorily explains the relative values of SSRS intensities ISSRS∝eΓSSRS obtained in the different RPP configurations, where ISSRSRPP2P≪ISSRSRPP1≲ISSRSRPP2S. Approximating the laser intensity as constant across the spot (≈FWHM) and applying again the model of Mostrom and Kaufmann,3 we obtain gains for λ_SSRS = 700 nm of Γ_SSRS ≈ 6, 8.7, and 9.4 for RPP2-P, RPP1, and RPP2-S spots, respectively, resulting in I_SSRS enhancements of ≈15 and ≈2, going from the smaller to the larger FHWM spot. This is consistent with the experimental values.

A more detailed calculation of the gain could be obtained by modeling the ray-tracing of SSRS light in the time-varying 2D density maps. It is worth noting that the SSRS intensity observed for the RPP2-S shots cannot be explained without accounting for the plasma curvature, unless a mechanism of saturation is active. We also remark that the above interaction conditions imply that eigenmodes, i.e., absolute SSRS solutions, cannot grow in this experiment. According to Ref. 3, in fact, these modes prevail for times longer than t_s and require a laser beam larger than 2L_res ≈ 240 μm.

The limiting factors of SSRS growth discussed above suggest also a plausible explanation for the confinement of SSRS to densities higher than 0.13n_c in the RPP2-P case. Since the resonance L_res increases toward lower plasma densities, rising from ≈80 μm for n_e = 0.16n_c to ≈190 μm for n_e = 0.08n_c, the size of the beam (D = 40 μm) will be a stronger limiting factor at lower densities, resulting in a negligible level of gain.

Unlike SSRS, the shape of the focal spot does not affect the growth of BSRS. FAB calorimetry for RPP2 shots on thick targets yielded a BSRS reflectivity of ∼(0.025 ± 0.005)%, which is comparable to the value obtained with RPP1. Consistently, similar BSRS intensities were measured by the FAB streak camera, with average values differing by 50% in the two irradiation conditions, which is less than the shot-to-shot reproducibility. This result is expected, since BSRS grows convectively in the backward direction, and the gain is therefore not affected by laser spot size. According to the Rosenbluth model,26 the linear BSRS gain Γ_BSRS ∝ L_nI. Small variations of total BSRS reflectivity in RPP1 and RPP2 irradiation can therefore be produced by the different intensity profiles of the laser beam in the two cases.

Confirmation of the different roles played by L_n in the growth of SSRS and BSRS was obtained from the shots on thin foil targets of different thickness. A typical angularly resolved spectrum from a 2 μm target irradiated using RPP1 is shown in Fig. 8, while the scattered energies at different angles are plotted in Fig. 4(d). It is interesting to note that the ω₀/2 signal is still visible in the spectrum; this is confirmed by the FAB streak camera, revealing that TPD disappears at a time 100–150 ps after the laser peak, different from the profile reported in Fig. 5(b). Hydrodynamic simulations show in fact that the peak electron density falls below n_c before the laser peak, and below n_c/4 ∼100 ps after the laser peak, as shown in Fig. 2(a). n-BSRS is again observed in Fig. 8 in both the P- and S-planes with comparable intensity; the signal, however, is much higher than that obtained on thick targets (Fig. 3). By comparing the spectra obtained for different foil targets in RPP1 shots, we observe that the n-BSRS intensity falls with increasing target thickness, approaching that measured for thick targets [Fig. 7(b)]. Consistently, the pyroelectric detector in the FAB cone measured an average BSRS reflectivity with RPP1 of (0.14 ± 0.01)%, (0.08 ± 0.01)%, and (0.015 ± 0.010)% for 2, 3, and 5 μm foils, respectively, compared with (0.020 ± 0.005)% obtained for thick targets [green triangles in Fig. 7(b)]. A similar trend was obtained by using RPP2, where calorimetry of BSRS showed a fall by a factor ∼60 on passing from 2 μm foils to thick targets. The enhancement of n-BSRS in foil targets can be also observed in Figs. 4(b) and 4(d), reporting the angularly resolved scattered energies measured in RPP2-P and RPP2-S shots on 2 μm targets. All the curves plotted from thin foils, in fact, exhibit a decreasing trend of scattered energy vs angle, implying that n-BSRS, stronger at low polar angles, dominates over SSRS; in the S-plane, however, the scattered energy remains higher at larger angles, owing to the contribution of SSRS. The trends described above confirm the strong dependence of BSRS and n-BSRS on L_n, as expected from the Rosenbluth theory.

The SSRS spectrum is also visible in Fig. 8 and clearly distinguishable in the S-plane, showing features similar to those in Fig. 3. The total intensity is shown in Fig. 7(b) for different targets. Despite the poor reproducibility, the statistical average of SSRS intensity is comparable for all the targets, confirming that its growth is not affected by the density scale length. As discussed above, this follows from the fact that the effective resonance length is dictated by the laser spot size or by the plasma curvature. Similar information on SSRS can be obtained by comparing the energy curves in the S-planes in Figs. 4(d) and 4(f) with those in Figs. 4(c) and 4(e), corresponding to thick targets. It is evident that the maximum value of the scattered energy is similar in the two cases; however, for thick targets, the curve decreases at smaller angles, owing to the lower contribution of n-BSRS.

In conclusion, angularly and spectrally resolved measurements of scattered light have allowed us to obtain a detailed characterization of BSRS, n-BSRS, SSRS, and TPD at laser intensities relevant to the shock ignition scheme for ICF. Variation of focal spot geometry and target thickness have allowed investigation of the dependence of BSRS and SSRS on the spot features and on the electron density scale length of the plasma. While BSRS (and n-BSRS) scales with L_n, as expected from linear theory, SSRS has been shown to be insensitive to L_n and to increase with the extent of the focal spot in the S-direction, which is the most favorable direction of growth. Consistently, the measured SSRS signal is much stronger in the S-plane than in the P-plane. This implies that the resonance length of convective SSRS is limited by the size of the laser spot or by the plasma curvature, and is therefore smaller than the value calculated for a planar infinite plasma, which scales with L_n.

It is worth remarking that both the density scale length and the focal spot size are here much smaller than the values envisaged in realistic ICF conditions. As BSRS reflectivity is expected to boost to tens of percent in plasma coronas of hundreds of micrometers length at intensities ∼10¹⁶ W/cm², SSRS can also become much stronger for larger laser spots, which are needed to obtain homogeneous laser irradiation of the fuel capsule. The results presented above suggest that estimation of SSRS reflectivity must account for realistic irradiation conditions and 2D plasma profiles; this holds in particular at lower plasma densities, where the size of the beam and the radius of curvature of the plasma, both of the order of a millimeter, become comparable to the resonance length, thus strongly limiting SSRS growth.

Acknowledgment. We would like to thank J. Myatt, P. Michel, and D. Carleton for useful discussions and valuable suggestions. We would like to acknowledge financial support from the LASERLAB-EUROPE Access to Research Infrastructure Activity (Application No. 23068). This work has also been carried out within the framework of EUROfusion Enabling Research Projects AWP21-ENR-01-CEA-02 and AWP24-ENR-IFE-02-CEA-02, and has received funding from Euratom Research and Training Programme 2021–2025 under Grant No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. We also acknowledge the NextGeneration EU Integrated Infrastructure I-PHOQS—Initiative in Photonic and Quantum Sciences (CUP B53C22001750006, ID D2B8D520, and IR0000016). This work is also supported by the Ministry of Youth and Sports of the Czech Republic [Project No. LM2023068 (PALS RI)] and by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant Nos. XDA25030200 and XDA25010100). This work has been supported by COST (European Cooperation in Science and Technology) through Action CA21128 PROBONO (PROton BOron Nuclear Fusion: from energy production to medical applicatiOns).