Stimulated Brillouin scattering dependence on polarization state, speckle shape, and polarization smoothing implementation

High power laser facilities such as the National Ignition Facility (NIF) in United States,¹ the SG-III laser in China,² or the Laser Megajoule (LMJ) in France³ are designed and built to trigger fusion reactions by inertial confinement (the so-called Inertial Confinement Fusion [ICF]). For this purpose, they use an indirect drive scheme in which the deuterium-tritium target is placed in a gold hohlraum illuminated by multiple laser beams.^4–7 In this configuration, both the gas present inside the hohlraum and the gold wall are ionized during the laser/target interaction. When laser beams propagate through the laser-generated plasma, coherent energy coupling processes appear and Laser-Plasma Instabilities (LPI) can occur such as Stimulated Raman Scattering, Stimulated Brillouin Scattering (SBS), Cross Beam Energy Transfer, two-plasmon decay, and self-focusing.^8–11 LPI are deleterious to the success of the experiment¹² since the propagation of the laser beam in the plasma is not controlled resulting in unwanted location of the laser deposited energy.¹³ Moreover, a part of the laser energy is backscattered with two consequences: (i) this backscattered energy does not contribute to the target implosion, and therefore, degrades the experimental yield,¹⁴ (ii) the backscattered energy can also trigger damages in the laser system.¹⁵ 

Laser beam smoothing techniques are commonly implemented in high-energy ICF laser systems to limit LPI. Their role is to reduce the coherence between the laser beam and the plasma, thereby reducing the coherent coupling energetic processes triggered by LPI. In direct drive scheme, the smoothing techniques are essential for preventing the hydrodynamic instabilities.¹⁶ Most common beam smoothing techniques¹⁷ are the Smoothing by Spectral Dispersion,¹⁸ the color beam smoothing¹⁹ or the Polarization Smoothing (PS).^20–24 They are complementary and can sometimes be combined.^22,25,26 Overall, the implementation of laser beam smoothing techniques is generally limited by technological issues. They are also often implemented at the expense of other laser performances such as the maximal energy of the laser system provided to the target. It is particularly true for the original PS scheme (later called intra-beam PS, in this paper) where a prismatic birefringent crystal creates two orthogonal states of polarization with a small angular separation to generate the two uncorrelated speckle patterns.^20–22 For the 40  $×$ 40 cm² laser beam square sections used in NIF and LMJ, KDP/DKDP (potassium dihydrogen phosphate/deuterated potassium dihydrogen phosphate) are the only crystals available to achieve this task. When used at high intensity at the wavelength of 351 nm, Transverse Stimulated Raman Scattering (TSRS) occurs in the KDP or DKDP^27,28 and can damage crystals on their edges.²⁹ This limits the maximal energy/intensity of the laser system. Alternative PS implementations can be used:³⁰ engineering the crystal orientation to limit TSRS,³¹ positioning the crystal before the frequency conversion at the wavelength of 1053 nm, to decrease the gain coefficient of TSRS, or using liquid crystal optical devices.³² In particular, the initial intra-beam PS implementation may be replaced by an intra-quadruplet PS configuration where the linear polarization state is rotated for two beams of the quad with respect to the two others.^26,33 In this scheme, the polarization rotator crystal can be placed at the wavelength of 1053 nm but induces losses due to its absorption at this wavelength. However, the limitation induced by TSRS is suppressed. For LMJ,³ the Frequency Conversion and Focusing System (FCFS) used does not offer this degree of freedom. The FCFS is designed to optimize and enhance the spectral acceptance of the frequency conversion thanks to the chromatic dispersion introduced by the beam steering gratings.³⁴ However, the dispersion direction imposes the linear polarization direction. Therefore, the polarization must be rotated at the wavelength of 351 nm. Then, an alternative solution was developed to avoid TSRS in DKDP crystal using a metamaterial quarter-wave plate operating at the wavelength of 351 nm.³⁵ This full-silica metamaterial wave plate converts the linear polarization into a circular polarization and provides two orthogonal left and right circular polarization states. The circular polarization also reduces the self-focusing Kerr effect in thick optics and significantly increases the filamentation threshold in the final optics assembly.³⁶ 

Implementation of intra-quadruplet PS on high energy density laser facilities then raises two questions: (i) Does the state of polarization (linear or circular) play a role on the LPI development? (ii) Does the statistical average speckle shape which may be modified by the intra-quadruplet PS implementation, play a role in the LPI growth? We address these two questions through an experimental approach. We focus on the reflectivity of the Stimulated Brillouin Scattering, which is one of the most significant instabilities occurring in ICF experiments. In addition, we also study the quality of laser transmission in the plasma.

The article is organized as follows: in Sec. II we present the experimental setup, the tested laser configurations, the measured and simulated plasma conditions. In Sec. III, the SBS reflectivity and transmission measurements are presented while numerical results for SBS reflectivity are shown in Sec. IV. In Sec. V, we discuss and characterize the reflectivity and transmission with only one laser parameter. Section VI concludes the article.

A laser beam interacts with a preformed plasma. The evolution of both the stimulated Brillouin backscattering reflectivity and the transmitted laser beam are measured with respect to a variation of the incoming laser beam intensity. Different laser beam configurations are investigated with the same phase plate component. Since the point spread function size is 20 times smaller than the size of the focal spot envelope,^37–39 the size and the shape of the focal spot envelope, which is positioned at the center of created plasma, remain unchanged throughout the experimental campaign. For each different configuration, we modified the state of polarization (linear, circular), the beam smoothing scheme used (no beam smoothing, intra-beam PS,^20,21 or intra-doublet PS^26,33) the near field beam geometry (square or rectangular aperture) and consequently the statistical average shapes and dimensions of speckle spots in the far field. The tested configurations are summarized in Table I.

The experiment was carried out at the LULI Laboratory (Ecole Polytechnique, Palaiseau, France) using the LULI2000 laser facility.⁴⁰ The setup is shown in Fig. 1. A first beam, denoted as the heating beam, with an energy of 250 J and a 1.5 ns square pulse duration (FWHM) at $2ω$ (⁠ $λh=526.5$ nm), is used to preform the plasma by ionizing the nitrogen gas. It is focused by an $f/9$ lens with a phase plate to obtain a 840 μm top-hat diameter far-field focal spot. The resulting intensity is about 3–5  $×1013$ W/cm². A second beam, denoted as the interaction beam, has a 1.1 ns square pulse duration (FWHM) at $1ω$ (⁠ $λ0=1053$ nm) and is focused by a 400 mm focal length lens preceded by a phase plate to obtain a 295 μm top-hat diameter far-field focal spot. Note that the size of the near-field of this beam is reduced to a square of $a×a=90×90$ mm² due to the dimensions of the various polarizing optical elements, which limits the energy to a maximum of 100 J. The gas jet is obtained with a supersonic nozzle with a 1 mm diameter aperture. The estimation of the plasma density is described in Sec. II B. The interaction beam is fired into the plasma at 50° with respect to the heating beam and 1.25 ns after.

The near-field geometry of the interaction beam was modified as follows. Serrated apertures were inserted at the output of the laser front-end section to obtain either a square shape of the full aperture size or a single rectangular aperture (⁠ $a/2×a≈42.5×90$ mm²). A $5×100$ mm² absorbing glass in front of the focusing lens produces two rectangular (⁠ $a/2×a≈42.5×90$ mm²) apertures from the square aperture.

The polarization state (linear vertical, linear horizontal, circular) was modified using a DKDP half wave-plate or a quarter wave-plate placed in the near field. For configuration without smoothing, a fused silica plano-parallel plate with the same optical thickness as the DKDP half wave-plate was inserted in place of the crystal.

For the intra-beam PS configuration, PS was performed using a prismatic DKDP crystal, whose angle is 1 mrad, with its neutral axis placed at 45° of the incoming linear polarization. Cutting angle of the crystal ensures a separation of 50 μrad between the propagation directions of each of the two polarization states.

The temporal profiles and energies of both the heating beam at 2  $ω$ and the interaction beam at 1  $ω$ are characterized at each shot using calibrated photodiodes and calorimeters.

Both focal spots are characterized with front-end low energy beams using a CCD camera. The diameter is estimated by measuring the encircled energy. The heating beam focal spot diameter is estimated to 840 μm with 85% of laser energy. The interaction beam focal spot measurement is slightly different depending on the configuration (square beam, rectangular beam, PS) but is mainly governed by the phase plate. We have estimated a mean value of 295 μm diameter with 60% of laser energy.

The uncertainty of the evaluation of the interaction beam intensities is estimated to be $±30$%. However, this uncertainty consists of a measurement bias identical for all shots and all configurations, and an instability of shot-to-shot measurements, which is less than $±10$%. However, the regularity of the five reflectivity curves shown in Sec III supports our view that the positioning of the points on the curves is probably better than the assessment of measurement uncertainty.

Two sets of diagnostics on the interaction beam were used to perform transmission and reflectivity measurements. SBS power was measured in the focusing cone (⁠ $f/2.5$⁠). We obtained for each shot the time-resolved spectrum, using a spectrometer coupled with a streak camera, and the time-resolved reflectivity using a fast photodiode coupled with a 8 GHz oscilloscope. The time-resolved transmission is measured, by collecting the energy through an $f/2.8$ lens behind the plasma, by using also a fast photodiode coupled with a 8 GHz oscilloscope. A Spectralon screen, located just in front of the transmission lens, with a square, or rectangular, aperture respecting the dimensions of the interaction beam in the vacuum, has been used to evaluate the energy transmitted through the plasma and scattered out of the laser cone (Fig. 1). The uncertainty of the evaluation of transmissions as well as reflectivities is estimated to be less than $±10$ %. Shot-to-shot measurement instability is the main contributor here. Only the linear polarization configuration with the rectangular near field beam shape has a measurement bias of transmission of 20 % compared to other configurations since a specific definition of the Spectralon aperture was needed inducing possible uncertainty in the position of the aperture.

Knowledge of the plasma parameters is essential for the analysis of the experimental results. Since it was not possible to implement all plasma diagnostics on each shot, plasma characterization is partly performed by dedicated shots and compared to simulations.

Regarding Thomson scattering, the $2ω$ heating beam is used as self-scattering probe beam [Fig. 3(a)]. The light is collected by an $f/4$ lens at an angle of 165° with respect to the heating beam direction. Because of the large focal spot diameter of this $2ω$ beam and of the large Thomson scattering angle, the scattered light emerges from almost the entire profile of the gas jet. Moreover, with this geometry, electron fluctuations having typically $kpλD≈0.27$ are probed, which explains the smooth Thomson spectral shape shown in Fig. 3(b) (⁠ $kp$ is the wave number of the electron plasma waves and $λD$ is the electron Debye length). However, the maximum wavelength of the scattered light is directly related to the electron density at the top of the plasma profile. Note that to be more representative of the interaction shots, the 1  $ω$ interaction beam is also fired for this plasma characterization shot. By assuming an electron temperature of 0.27 keV, taking into account the interaction beam energy (see above), analysis of Fig. 3(b) shows that the typical maximum electron density $ne$ at the top profile is $9.5×1019$ $cm −3$⁠, or 0.095  $nc$⁠, where $nc$ is the critical density for the 1  $ω$ interaction beam.

To get additional information on the plasma, modeling was also carried out hydrodynamic simulations with an upgraded version of the three-dimensional (3D) HERA code.^44–46 This code solves the laser propagation and absorption in an under-dense sub-quarter critical plasma, and the laser's Stimulated Brillouin Scattering, which will be discussed in Sec. IV, together with the hydrodynamic evolution of the laser-heated plasma. The laser and B-SBS electromagnetic waves are coupled to an ion acoustic wave, each wave being described by a paraxial operator. This three-wave system is then coupled to a full hydrodynamic solver. A similar system is solved in pF3D.⁴⁷ The plasma is treated as a two-temperature fluid, with ions and electrons following a perfect gas equation of state, assuming full ionization of the nitrogen gas by the 250 J heater beam. The ion and electron heat fluxes are calculated using Spitzer–Härm conductivity with a 15% flux limiter. It appears that the plasma dynamics is not too sensitive to the flux limiter due to smooth temperature gradients. We performed simulations using a 10% flux limited and no limitation with the same results. This allows to model both the plasma evolution under thermal and ponderomotive pressures, and the propagation of the laser speckles, taking into account the phase of the electric field. Because we use a frequency dependent paraxial solver, we simulate the whole experiment by performing two successive simulations: first, we simulate the $2ω$ heating beam-plasma interaction up to the time corresponding to the rise of the $1ω$ interaction beam, using the measured gas jet density profile. This first simulation provides the initial conditions for the $1ω$ interaction beam-plasma simulation. Note that we use the experimental pulse powers for each beam. The main advantage of using HERA being its ability to describe the plasma heating of a speckled laser beam, there is still some drawbacks, such as the plasma treatment with a fluid and the fact that both beams have the same angular incidence. However, this approach is a posteriori justified by a comparison with Troll simulations⁴⁸ and experimental measurements. Figure 4 shows the temperature and density evolution taken at the center of the interaction region, during the heating and interaction stages.

A comparison of the plasma parameters obtained from the analysis of the experimental measurements and the post-shot simulations, respectively, gives rather good agreements, and therefore, enables us to know the properties and state of the plasma used in this experimental campaign. In summary, we find that at the beginning of the interaction beam stage, the plasma is a fully ionized nitrogen plasma with an initial peak electron density of the order of 0.08  $nc$⁠, an ion temperature of the order of 80 eV and an electron temperature of the order of 200–300 eV, depending on the energy of the interaction beam.

Figure 5 shows the SBS reflectivity, defined as the ratio between the B-SBS energy and the incident interaction beam energy, the laser transmission, defined as the ratio between the energy of the transmitted beam and the incident interaction beam energy, in and out the laser cone, as a function of the mean laser intensity for the different laser configurations investigated in this experiment. For each configuration, an increase in the laser intensity leads to an increase in reflectivity up to saturation combined with the decrease in the beam-energy transmission fraction in the laser cone, and a slight increase in the transmission ratio outside the laser cone (i.e., a greater laser beam spray).

In Fig. 5(a), in the configurations where the interaction beam is linearly (red curve, circle dots) or circularly (green curve, triangle dots) polarized, we observe that the reflectivity curves are almost superimposed. This is also the case of the transmission curves, inside and outside the laser cone, [Figs. 5(b) and 5(c)]. This is an experimental indication that the SBS reflectivity and transmission do not depend on the polarization state (linear or circular) of the interaction beam.

However, in Fig. 5(a), when the interaction near-field beam features a rectangular shape (half of the square beam), the reflectivity curve (blue, square dots) is shifted to lower intensities compared to the reflectivity curve obtained when the interaction near-field beam is square-shaped (red, circle dots). In addition, the transmission inside the laser cone (respectively, outside the laser cone) is much lower (respectively, higher) when the interaction near-field beam shape is rectangular compared to the square case. This is an experimental indication that the shape and the area of the near-field beamlet, like those of the speckle spots, do have an impact on the mitigation of SBS and on the control of the transmission of the laser beam through the plasma.

The comparison of reflectivity and transmission curves displayed in Fig. 5 shows that the polarization smoothing (purple and orange curves) leads to a decrease in reflectivity and to an increase in the transmission in the laser cone. This result was expected and has been observed in previous simulations^21,49,50 and experiments.^22,23,49,50 However, we observe in this experiment that the reflectivity and transmission curves obtained for the intra-beam or intra-doublet PS (see Table I) do not overlap: the intra-beam PS is more efficient regarding these two quantities.

As indicated in Sec. II B, we use the 3D HERA code both for determining the plasma parameters produced by the $2ω$ heating beam and for calculating the $1ω$ interaction beam backscattering. The whole simulation of one experimental shot requires two successive simulations on the same computational grid as explained earlier, first at $2ω$ and then at $1ω$ without any temporal overlap. It requires also an adaptation of the beam geometry. We remind that experimentally, the two laser beam directions make a 50° angle, but the HERA laser beam transverse boundary condition requires small angles due to its paraxial solver. Noting that the heater beam is wider than the gas jet diameter, the gas jet is almost isotropically heated. Combining those two facts, namely, that the plasma parameters are essentially isotropic and that the laser pulses are well temporally separated, we choose to inject the interaction beam in the same direction than the heater beam. In the following, we assume that the heating phase is always the same and we solely vary the interaction beam configuration and energy.

We simulate the $1ω$ interaction beam propagation for various intensities and the five configurations given in Table I. To make our simulations as realistic as possible, e.g., having the right laser intensity on target, it is necessary to adapt the simulated laser beam geometry. In particular, experimental measurements of the interaction beam focal spot shape indicate that 60% of the laser energy is contained in a well-defined $ϕ exp =295$ μm diameter round spot size, and 40% are lost in the focal spot wings. The experimental laser intensity is then defined by the ratio of the average power to the surface given by $πϕ exp 2/4$⁠. We choose to model the interaction beam accordingly in our simulations and focus on the B-SBS energy.

Calculated SBS reflectivity—the ratio in the cone outgoing B-SBS energy to the incoming energy—are presented in Fig. 6(a) for all configurations considered in the experiment and different laser intensities (energies). We see the same ordering as in the experiment: the instability threshold increases from linear polarization configuration with the rectangular near field beam shape (blue squares) to the intra-beam PS (orange squares). We validate the equivalence between linear polarization (red dots) and circular polarization (green dots). We also confirm the greater efficiency of intra-beam PS over intra-doublet PS (purple triangles). Those data are time-integrated over the full pulse duration and agree quantitatively well with the experimental ones.

It is interesting to compare the temporal evolution of the calculated B-SBS compared to the experimental ones. We show a typical B-SBS power calculated for a 52.7 J interaction beam Fig. 6(d) over its 1.1 ns full pulse duration, in dashed red line, compared to the shot experimental reflectivity, in solid red line (note that to improve the readability of the figure, both SBS power, numerical and experimental, are multiplied by ten [ $×10$]). The temporal evolution of the B-SBS power agrees with the experimental ones over all the pulse duration. The B-SBS power trend reflects the effect of the laser drive pulse shape, see black curve in Fig. 6(b), and of the plasma dynamics, that both determine the plasma temperature, laser absorption and density decrease due to the plasma expansion. Note also that, out of the laser cone, simulations indicate an amount of B-SBS energy comparable or greater than the ones inside the laser cone (not shown in Fig. 6). This will impact the energy balance when compared to experimental results. Unfortunately, there were no such measurements on the experimental platform.

Although our primary objective is the study of B-SBS, HERA simulations give also laser transmission data. We show time-resolved transmission data in Fig. 6(d) in the aperture angle of the laser: the solid blue line corresponds to the experimental measurement and the dashed blue line corresponds to the HERA results. We see a net discrepancy, the calculated transmission being larger than the experimental one. We observe a systematic overestimation of the calculated transmitted energy compared to experiments in all simulations. Quantitatively, simulations give 75%–90% full transmission—inside [Fig. 6(b)] and outside [Fig. 6(c)] the aperture angle—whereas experiments show 55%–70% [Figs. 5(b) and 5(c)]. Looking more in detail, the fraction of measured transmission outside the aperture angle represents 20%–30% in the experiment, when it represents 5%–20% in the simulations. Comparing those numbers is difficult because our simulations only consider the central part of the real laser beam when transmission measurements correspond to the whole laser beam. However, the laser absorption seems to give the right temperatures and B-SBS powers are well reproduced. Finally, the discrepancy may be attributed to out of the aperture B-SBS and laser refraction on his path through the whole gas jet which is numerically underestimated due to the absence of large focal spot wings is the simulation. Because B-SBS develops on the gas jet front, it is not affected.

Actually, the experimental data [Fig. 5(c)] only show a slight dependence of the interaction beam intensity on the transmission outside the aperture angle. Moreover, as indicated above, the fraction of simulated transmitted light out of the laser cone is relatively small (below 20%). Both observations are an indication that the role of forward laser plasma instabilities (such as Forward Stimulated Brillouin Scattering [F-SBS] or self-focusing) remains rather limited in the B-SBS development. The absence of important role of self-focusing is analytically confirmed by using the filamentation figure of merit (FFOM) mentioned in Refs. 51 and 52. If we look at the plasma parameters at the beginning of the plateau of the interaction pulse (⁠ $t∼1.8 ns$⁠), we can see from the results of the hydrodynamics simulations shown in Fig. 4 that $ne/nc∼6%$⁠, $Te∼270 eV$⁠, and $Ti∼80 eV$⁠. The value of FFOM for these parameters is $∼0.68$ for $〈I〉=3×1013 W/cm2$ (which appears to be the B-SBS threshold). This indicates that we are slightly below the self-focusing threshold. F-SBS is a better candidate to provoke angular spreading since its threshold is much lower than self-focusing one in our experimental conditions. However, as mentioned in the literature (for example in Fig. 1 of Ref. 51, or in more recent studies such as Ref. 53), F-SBS needs an important propagation distance to grow (typically several millimeters). And, as shown in Fig. 4(d), the plasma length is only a few speckle lengths in our study (and even less if we consider the length of the plasma with $ne/nc>5%$⁠). We believe that this is the reason why the angular spreading produced by F-SBS remains rather limited and does not dramatically change the sizes of the speckles. This means that F-SBS (and maybe also self-focusing), even developing, cannot be the only cause of the observed shift in the experimental reflectivity curves between the different laser configurations.

In this section, we propose an interpretation of the experimental results based on the speckle features. It is mainly an extension of previous works^54–60 with the addition of the contribution of the statistical average shape of the speckles. Although simple, we will show that the model we derive is in very good agreement with the experimental data.

The reflectivity curves shown in Fig. 5(a) clearly demonstrate that the SBS threshold depends on the used smoothing scheme. In addition, we know that the laser configuration induces the properties of the optical field in the focal spot volume, in particular the intensity distribution of the hot spots.

So, $Ahs$ is a nondimensionalized parameter depending only on the average shape of the hot spots, and thus, on the shape of the near field of the interaction beam because the variables $λ0$⁠, $Lhs$⁠, et $Shs$ are linked by diffraction.⁶⁴  $Ahs$ neither depends on the f-number, nor on the average size of the hot spots, nor on the size of the focal spot, nor on the laser wavelength. More precisely, if $Ahs$ depends on these parameters, it is a dependence that links these parameters.

Note that Eq. (5) does not establish an explicit expression for the SBS response, but establishes a single variable that describes the laser parameters on which this response depends. This variable allows for the decoupling of independent contributions related to the laser architecture, and highlights the influence of beam geometry. Considering the aspect shape ratio constitutes a simple extension of previous works.^54–60 

We would like to emphasize that the Eq. (5) treats the $RSBS$ as the response of a speckle pattern described by a statistical distribution characterized by its mean parameters, and not as the response of the average hotspot.

Reflectivity and transmission measured in the experiment are now represented in Fig. 7 with respect to the $Ahs Phs$ parameter normalized by $(Ahs/Nm)ref$⁠. The scale of the abscissas becomes a relative quantity that is uniquely accessible in this proposed model based on laser variables, but is useful for comparing different configurations. We can see that the SBS reflectivity and transmission in the cone curves are almost superimposed. The configuration with a rectangular near field and one polarization (blue curve, square dots) is the only one to deviate from this finding for the transmission measurements due to a singular measurement bias.

Note that it is possible to superimpose the reflectivity curves obtained in the reference and the intra-beam PS configurations, by shifting the intra-beam PS configuration curve by a factor $m=2$⁠, in accordance with the experimental and simulated results of references Refs. 49 and 50.

In the same way, we apply the parameter scaling to the simulated SBS reflectivity and transmission (Fig. 8). We obtain a very good agreement in reproducing the reflectivity scaling for all configurations even if the plasma characteristics were not the same for points of same reflectivity [see the electron temperature in Figs. 2(b) and 4(c)]. This also works for the simulated transmissions.

This overlap of reflectivity and transmission curves shows that the Brillouin reflectivity and the transmission in the cone are driven by the quantity $Ahs Phs=AhsPbN m$⁠. Then, the different laser configurations tested during this experiment, and even those that were not, can be quantitatively compared based solely on the value of the three parameters: $Ahs$⁠, N, and m. For example, the fact that, for a fixed total laser power $Pb$⁠, the linear polarization configuration where the shape of the near-field interaction beam is rectangular (blue curve, square dots) gives rise to higher reflectivity and lower transmission ratio than the reference configuration is because the shape aspect ratio of the speckles has changed for the better (⁠ $Ahs/(Ahs)ref=0.68$⁠) but the average number of hot spots over which laser energy is distributed has been divided by a factor of 2 (⁠ $N/Nref=1/2$⁠). Similarly, the fact that the intra-beam or intra-doublet PS are not equivalent is explained by the fact that, despite $m=2$ in both cases, the average number of hot spots N and the shape aspect ratio $Ahs$ are not the same in both configurations (see Table II).

Overall, these results demonstrate that, to increase the SBS threshold and to limit SBS reflectivity, it is favorable to retain a configuration increasing the statistical average number of hot spots N, using two polarization states $m=2$ and a geometry of the near field interaction beam as far as possible from circular (to get a small shape aspect ratio $Ahs$⁠). In other words, as an example, it is worth noting that for the same near field beam area (which was not the case in our experiments), a rectangular laser aperture reduces B-SBS as compared to a square aperture.

Therefore, when designing a future laser facility for Inertial Fusion Energy (IFE), it will be interesting to consider the near field beam geometry to limit LPI.

We have experimentally demonstrated and numerically confirmed that under the plasma conditions considered in this experiment, the linear or circular state of polarization has no effect on SBS reflectivity and transmission. This first conclusion offers flexibility when it comes to implement polarization smoothing on a high power laser facility,³⁵ or circular polarization to limit the Kerr effect at the wavelength of 351 nm in the final optics.³⁶ 

We have shown that it is possible to sort the different experimental conditions with respect to SBS reflectivity and transmission. In particular, the intra-beam PS is the most efficient configuration for the laser-plasma interaction, favoring a low SBS and a high transmission. The intra-doublet configuration turns out to be less efficient than the intra-beam PS configuration.

We have also proposed a laser parameter taking into account the number of independent polarization states and associating the statistical average power of the hot spots with a dimensionless quantity representing the shape aspect ratio of the hot spots. With this formulation, under the plasma conditions considered in this experiment, we are able to compare the various laser configurations tested in the experimental campaign and even those that were not. However, we must bear in mind that this model is relatively simple and assumes in particular, that the number and the sizes of hot spots are not modified by their propagation in the plasma. In some real laser-plasma interaction experiment setups, self-focusing or F-SBS can clearly modify speckle sizes in the plasma.

The statistical average shape aspect ratio of the speckle spots and their statistical average number in the focal spot do have an impact on the SBS and the plasma transmission. To design a polarization smoothing implementation for a high power laser facility, it is beneficial to choose a design that decreases the shape aspect ratio of the speckle spots and increases their statistical average number in the focal spot.

These findings bring a solid baseline to propose new laser beam smoothing techniques for high-power lasers, in particular in the context of potential IFE reactors.^71–73 

This work was granted access to the HPC resources of TGCC under the allocations A00030506129 and A0150513036 made by GENCI (Grand Equipement National de Calcul Intensif) and to the TERA-1000-2 supercomputer of CEA-DAM. We acknowledge the beneficial support from LULI technical staff during these experiments and thank N. Bonod for fruitful guidance in the writing of this manuscript.

The authors have no conflicts to disclose.

J. G. Moreau: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). N. Blanchot: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (lead); Project administration (equal); Resources (equal); Validation (equal); Writing – original draft (equal). C. Rousseaux: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (equal); Resources (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). S. D. Baton: Data curation (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). D. Penninckx: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Methodology (equal); Project administration (lead); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). A. Fusaro: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (lead); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). P. Loiseau: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). R. Collin: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (supporting); Software (equal); Validation (supporting). G. Riazuelo: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Writing – review & editing (equal). P.-E. Masson-Laborde: Conceptualization (equal); Formal analysis (equal); Methodology (supporting); Software (equal); Validation (equal); Writing – review & editing (equal). J. P. Zou: Resources (supporting). L. Lancia: Investigation (supporting). C. Rouyer: Conceptualization (equal); Formal analysis (lead); Methodology (equal); Validation (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (lead). C. Maunier: Resources (supporting). X. Ribeyre: Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). H. Coic: Resources (supporting). O. Selwa: Resources (supporting). J. Daurios: Resources (supporting). J. Neauport: Conceptualization (supporting); Funding acquisition (equal); Project administration (equal); Resources (supporting); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.