Introduction

In laser-driven inertial confinement fusion (ICF), laser or soft x rays generated in a hohlraum is used to drive the implosion of a spherical capsule containing deuterium and tritium to achieve thermal nuclear fusion ignition1,2,3. For a long time, the path to ignition on National Ignition Facility (NIF) were blocked by hydrodynamic instabilities4,5,6,7,8 and radiation drive asymmetries in cylindrical hohlraums9,10. Hydrodynamic instabilities can cause kinetic energy loss, ablator-fuel mixing and hot spot degradation in capsule implosions, reducing fusion efficiency, while drive asymmetries can lead to insufficient fuel compression and lower the energy conversion efficiency from kinetic energy of capsule to internal energy of hot spot. Although ignition has been achieved at NIF recently11,12,13, many issues are still not well understood. The issues of prime importance in cylindrical hohlraums, include the drive asymmetries and hohlraum energetics which are closely related to laser plasma instabilities (LPIs)14,15. For indirect drive ICF, stimulated Brillouin backscattering (SBS) and stimulated Raman backscattering (SRS) are the LPIs which may reduce the laser target coupling, preheat the fuel, and damage the optic elements16,17. The crossed-beam energy transfer (CBET)18,19 is another important LPI, which can affect the drive symmetry and plays a crucial role in achieving ignition at NIF11,12,13. However, because CBET in hohlraums is not fully understood, the symmetry tune of hohlraum drive is based on a dedicated experimental database20.

A great deal of efforts have been engaged in understanding LPIs21,22,23,24,25,26,27 and suppressing backscattering28,29,30,31. In indirect drive ICF experiments, laser beams arranged in cones enter into a hohlraum from both sides. It is generally assumed that the backscattering of laser beams in the same cone is identical, since these beams are incident at the same angle relative to the hohlraum axis and symmetrically distributed. As a result, only one suite of diagnostics is routinely placed on a laser beam in one cone for measuring backscattering. So far, there is no direct experimental evidence showing that the backscattering is not identical in the same cone. However, if backscattering in the same cone is not identical, it would result in the azimuthal drive asymmetry and degrade the implosion performance. In addition, the asymmetric backscattering also introduces obvious difficulty to obtain accurate backscattered energy in the experiment due to the limited diagnostic access. Therefore, the asymmetric backscattering of laser beams in the same cone would be an obstacle in understanding hohlraum energetics and may be a potential reason for the energy deficit32.

In the hohlraum experiment at SG-100kJ laser facility33,34, it is found that backscattering of laser beams in the same cone behave quite differently. The experiment results clearly show a close relation between the backscattering and the polarization of laser beams. Based on numerical simulations of backscattering and multi-beam CBET, it is revealed that the laser beams in the same cone obtain different power from their neighbors via CBET when their polarizations are not symmetrical in azimuthal direction, therefore resulting in different backscattering. Because multi-beam CBET process in hohlraum is rather complicated, the dependence of multi-beam CBET on laser polarization in real hohlraum plasma conditions is investigated in a specially designed CBET experiment, and validated with a numerical simulation of multi-beam CBET.

Results

Backscattering experiment

In the backscattering experiment at the SG-100kJ laser facility, 48 laser beams at a wavelength of 0.351 μm are injected into a cylindrical gold hohlraum from upper and lower hemispheres at angles of 28. 5∘, 35∘, 49. 5∘ and 55∘ with respect to the hohlraum axis. The two inner cones at 28. 5∘ and 35∘ have four beams. The two outer cones at 49. 5∘ and 55∘ have eight beams. Figure 1 presents a schematic view of the experiment and the laser pulse used to drive the hohlraum. In order to investigate whether the backscattering of laser beams in the same cone are identical or not, two sets of Full Aperture Backscatter Station (FABS) were installed on the beams A1 and A2 in the outer cone of 55∘ at lower hemisphere, as shown in Fig. 1.

Fig. 1: Sketch of the backscattering experiment setup and the laser pulse.
Fig. 1: Sketch of the backscattering experiment setup and the laser pulse.
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Red dotted line represents the power of heat beams of which the laser pulse before 1 ns is used to ablate the polyimide window over the laser entrance hole. Blue solid line represents the power of interaction beams. The beam incident angles from outer to inner are 55∘ (blue), 49.5∘ (cyan), 35∘ (pink) and 28.5∘ (red). The backscattered lights of Beams A1 and A2 are diagnosed by two sets of Full Aperture Backscatter Station (FABS).

Two cases are compared in the backscattering experiment. Figure 2a shows the geometry and polarization configurations of the 24 beams at the lower hemisphere in case I, where the polarization direction of A1 is along the tangential direction of the circle formed by the cone of 55∘, and the polarization direction of A2 is perpendicular to the tangential direction. In case II, the polarization direction of A1 is changed to be perpendicular to the tangential direction, as shown in Fig. 2c. Table 1 presents the measured backscattering energy of beams A1 and A2 in both cases, including SBS and SRS. The experimental results clearly show that the backscattering of laser beams are directly related to their polarizations. In case I, the backscattered energies of SBS and SRS of A1 are respectively 280 J and 34 J, which are larger than those of A2 (198 J of SBS and 30 J of SRS). In case II, the backscattered energies of SBS and SRS of A1 are respectively reduced to 71 J and 16 J, which notably lower than those of A2 (212 J of SBS and 30 J of SRS). In these two cases, regardless of whether the polarizations of A1 and A2 are the same or not, the backscattered energies of A1 and A2 are quite different. Furthermore, when the polarization of A1 is changed from case I to case II, its backscattered energies are greatly reduced, but those of A2 remain the same.

Fig. 2: Configuration of 24 beams at the lower hemisphere in the backscattering experiment.
Fig. 2: Configuration of 24 beams at the lower hemisphere in the backscattering experiment.
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a Arrangement of the 24 beams and their polarizations for case I from an upward perspective. b The sketch of polarization configuration of the beams in group 1 and group 2 for case I in the backscattering experiments. c Arrangement of the 24 beams and their polarizations for case II from an upward perspective. d The sketch of polarization configuration of the beams in group 1 and group 2 for case II in the backscattering experiments. The arrows indicate the direction of polarization projected onto the plane of laser entrance hole for each beam. The beam incident angles from outer to inner are 55∘ (blue), 49.5∘ (cyan), 35∘ (pink) and 28.5∘ (red).

Table 1 Energy of backscattered light measured by FABS in the backscattering experiment
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The experimental time-resolved spectra shown in Fig. 3 and the powers of SBS backscattered light shown in Fig. 4a clearly exhibit the difference between case I and case II for the beam A1. In case I, the SBS of A1 mainly occurs around 1 ns and in the interval of 3−4 ns. However, in case II, the SBS of A1 decreases remarkably in the interval of 3−4 ns. This is an explicit evidence of strong correlation between the backscattering and the polarization of A1. The spectra and power of the SBS backscattered light of A2 are almost the same in the two cases, indicating that the polarization change of A1 has no influence on A2. Besides, even when the polarizations of A1 and A2 are the same in case II, their time-resolved spectra and powers of the SBS backscattered light are still quite different.

Fig. 3: Measured SBS spectra in the backscattering experiment.
Fig. 3: Measured SBS spectra in the backscattering experiment.
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a Time-resolved spectra of SBS backscattered light for beam A1 in case I. b Time-resolved spectra of SBS backscattered light for beam A2 in case I. c Time-resolved spectra of SBS backscattered light for beam A1 in case II. d Time-resolved spectra of SBS backscattered light for beam A2 in case II. Green lines are temporally-integrated spectra, and white lines are the spectrally-integrated powers.

Fig. 4: Laser power and backscattering power.
Fig. 4: Laser power and backscattering power.
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a Time dependence of the power of SBS backscattered light obtained in the backscattering experiment. b Time dependence of the incident power of beams A1 and A2 in the experiments and the simulated power of beams A1 and A2 after CBET. In the two figures, navy blue solid lines are the results of beam A1 in case I, blue solid lines are the results of beam A1 in case II, yellow solid lines are the results of beam A2 in case I, and orange solid lines are the results of beam A2 in case II. Black dashed line is the incident power of beam A1, and grey dashed line is the incident power of beam A2.

Analysis of the backscattering experiment

The experimental results clearly show that the backscattering of A1 and A2 in the same cone are quite different from each other, no matter their polarizations are the same or not. It indicates that the polarization is not the direct cause of the different backscattering behaviors of laser beams in the same cone.

CBET between two laser beams has been found to be dependent on their polarizations35,36,37,38,39, and CBET can occur between laser beams with the same wavelength in flowing plasma9,40. In the backscattering experiment, there are many laser beams overlapping near the laser entrance hole (LEH), making it possible to excite CBET between these overlapping beams. Because the overlapping laser beams have different polarizations, it is difficult to predict the power change of A1 and A2 via CBET, even though the incident laser powers of A1 and A2 are the same. Therefore, the dependence of CBET on laser polarization may be the reason for the different backscattering behaviors of laser beams A1 and A2 in the same cone.

To validate our hypothesis, the backscattering experiments are simulated by using the radiative hydrodynamic code LARED-Integration41,42,43 and a 3-dimensional ray-tracing CBET postprocessing code. The simulation results show that the presence of plasma flows in the LEH region induces dramatic energy transfer between laser beams with the same wavelength. The energy transfer primarily occurs between the nearest beams. As shown in Fig. 2a and b, the energy transferred to A1 at 55∘ is dominantly taken from beam B1 at 28. 5∘, and two beams C1 and D1 of 49. 5∘, because these three beams have similar polarizations and possess overlapping volumes with A1 larger than other beams. These four beams are labeled as group 1 in Fig. 2b. The energy is transferred from the outer cone beams to the inner cone beams in the interval of 1−1.5 ns and it is transferred from other cones to the cone of 55∘ in the interval of 3−4 ns, as shown in Fig. 4b. The incident laser power is measured in the experiments, and the final power after CBET is obtained from the simulations. In case I, the polarization of A1 is close to its three neighbor beams B1, C1 and D1, as shown in Fig. 2a, b, while the polarization of A2 is only close to its two neighbor beams B2 and D2 but nearly perpendicular to C2. Compared with A2, A1 loses more energy to its three neighbor beams during time of 1−1.5 ns but obtains more energy during 3−4 ns via CBET. As a result, the power of SBS backscattered light of A1 is lower than A2 at the first peak but higher than A2 at the second peak, as shown in Fig. 4a. In case II, the polarization of A1 is nearly perpendicular to its three neighbor beams, as shown in Fig. 2c and d, so the energy transfer between A1 and the other three beams is greatly reduced compared to case I. However, the CBET between A2 and its two neighbor beams B2 and D2 is maintained at the same level as in case I. Therefore, compared to case I, the power of A1 after CBET is higher during 1−1.5 ns but much lower during 3−4 ns in case II. Since the power of A2 after CBET is almost the same in the two cases, the SBS backscattering of A2 is maintained at the same level. Therefore the process of CBET is determined by the beam polarization arrangement.

In Fig. 5, we present the simulated time-resolved spectra of SBS backscattered light by the HLIP code44,45 with the powers of A1 and A2 after CBET in the two cases shown in Fig. 4b. The simulation results are consistent with the experimental observations shown in Fig. 3. The experimental and simulation results demonstrate that the different backscattering of A1 and A2 are originated from the different levels of CBET, which is caused by the azimuthally asymmetric polarization arrangement.

Fig. 5: Simulated SBS spectra in the backscattering experiment.
Fig. 5: Simulated SBS spectra in the backscattering experiment.
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a The simulated time-resolved spectra of SBS backscattered light for beam A1 in case I. b The simulated time-resolved spectra of SBS backscattered light for beam A2 in case I. c The simulated time-resolved spectra of SBS backscattered light for beam A1 in case II. d The simulated time-resolved spectra of SBS backscattered light for beam A2 in case II. Green lines are temporally-integrated spectra, and white lines are the spectrally-integrated powers.

CBET experiment

To confirm our conclusion that the dependence of CBET on laser polarization is the reason for the different backscattering of laser beams in the same cone, another experiment was performed at the SG-100kJ laser facility to directly show the dependence of CBET in hohlraum on laser polarization and verify our simulation codes used to analyze the backscattering experiment. A short cylindrical gold hohlraum was specially designed for this experiment. To reproduce laser conditions near the LEH region, the hohlraum is driven by 24 laser beams from lower hemisphere, as shown in Fig. 6a. The outer cone beams are used to heat the hohlraum walls, while the inner cone beams pass through it. The beam B1 in the inner cone of 28.5∘ is used as a diagnostic beam. It passes through the hohlraum and ablates a gold plate placed above it. The power change of B1 due to CBET is determined by the x-ray radiation flux of the laser spot region on the gold plate which is measured by a Space-resolving Radiation Flux Detector (SRFD)46,47. The temporal profile of laser pulse is shown in Fig. 6a. To enable energy transfer from the outer cones to inner cones, the wavelength separation was \({\lambda }_{in}-{\lambda }_{out}=1.67\,{{\text{\AA}}}\) with λout = 0.351 μm for the outer cone beams.

Fig. 6: Sketch of the CBET experiment and beam configuration.
Fig. 6: Sketch of the CBET experiment and beam configuration.
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a The sketch map of the CBET experiment setup with the laser pulse. Blue solid line represents the power of inner beams and red solid line represents the power of outer beams. The x-ray radiation flux is measured by a Space-resolving Radiation Flux Detector (SRFD). b The arrangement of 24 beams and their polarizations at lower hemisphere for case IV. The arrows indicate the direction of polarization projected onto the plane of laser entrance hole for each beam. The incident angles from outer to inner cones are 55∘ (blue), 49.5∘ (cyan), 35∘ (pink) and 28.5∘ (red), respectively.

Two cases with different polarizations of B1 are compared in this experiment. In case III, the laser configuration and polarization arrangement of 24 beams of lower hemisphere are maintained the same as in the case I in the backscattering experiment shown in Fig. 2a. The polarization of B1 is along the tangent direction of the circle formed by the four beams of 28.5∘ cone. In case IV, the polarization of B1 is changed to be perpendicular to the tangent direction of that circle, as shown in Fig. 6b.

The experiment is designed and analyzed by using the radiation hydrodynamic code LARED-Integration and the 3-dimensional ray-tracing CBET postprocessing code. Prior to the experiment, the simulation predicts that beam B1 mainly obtains energy from its three neighbor beams A1, C1 and D1 via CBET in case III, because their polarization directions are close to B1 and the overlapping region between the four beams is the largest. The polarization state of laser beams in this experiment is almost not affected by CBET. By contrast, CBET between B1 and its three neighbor beams is greatly reduced in case IV since the initial polarization of B1 is nearly perpendicular to the other three beams. Therefore, the laser power of B1 is increased in case III but almost unchanged in case IV after passing through the hohlraum.

The experimental results agree with the theoretical predictions. As shown in Fig. 7b, the measured radiation flux in case III is much higher than that of case IV, although the incident laser power of B1 in both cases is almost the same, as shown in Fig. 7a. The laser power of B1 after passing through the hohlraum, obtained from post-shot simulations, in case III and IV is also shown in Fig. 7a. According to the simulation results, the laser energy of B1 is increased by about 34% in case III but only 2% in case IV via CBET. With the laser power of B1, the x-ray radiation flux, calculated by hydrodynamic simulations, agrees well with the experimental data, as shown in Fig. 7b. In addition, the backscattering energy of B1 is 143 J in case III, which is much higher than the 54 J in case IV. The backscattering results are also consistent with the reduction of transferred energy to B1 after the change of its initial polarization. Therefore, the results of CBET experiment definitely show the dependence of CBET in hohlraum on the laser polarization and verify our simulation codes.

Fig. 7: Laser power and x-ray radiation flux in the CBET experiment.
Fig. 7: Laser power and x-ray radiation flux in the CBET experiment.
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a The incident laser power of beam B1 (dotted lines) in cases III (navy blue) and IV (yellow) and its transmitted power after passing through hohlraum (solid lines) in cases III (blue) and IV (orange). In case III, beam B1 obtains laser power from its neighbor beams via CBET. b The dotted lines are the measured x-ray radiation flux in cases III (navy blue) and IV (yellow), and the solid lines are the simulation results obtained with the laser power of beam B1 after passing hohlraum in cases III (blue) and IV (orange).

Discussion

It is found that the laser beams in the same cone exhibit different backscattering in the experiments, and the backscattering of laser beams are closely related to their polarizations. No matter the polarizations of two laser beams in the same cone are similar or perpendicular to each other, their backscattering are quite different. Our numerical simulations reveal that dependence of CBET on laser polarization is the main reason for this particular phenomenon. To confirm our interpretation, a specially designed CBET experiment is performed at the SG-100kJ laser facility. It directly demonstrates the dependence of CBET in a multi-beam configuration in hohlraum plasma on laser polarization and verifies our simulation codes used in analyzing the experiments.

In principle, the dependence of CBET on polarization can be mitigated by using polarization smoothing (PS)48,49 on each laser beam. We performed a simulation with the CBET postprocessing code to investigate the effect of PS. In our simulation, each of the 24 beams at the lower hemispheres has the equal amplitude for its two polarization components. The simulation result shows that the power of inner beams after CBET is identical for the same cone, however, the power difference still exists for laser beams in the same outer cone due to the spatial position configuration of the 24 beams.

Our investigation indicates that it is important to set a symmetric polarization and position arrangement in the azimuthal direction of the target chamber to avoid the appearance of different backscattering of laser beams in the same cone. Otherwise, the inadequate backscattering diagnostics may lead to incorrect evaluation of the hohlraum energetics32,50,51 and the azimuthal drive asymmetry12.

Methods

Experiments

The backscattering experiment is conducted at SG-100kJ laser facility which consists of 48 laser beams. The beams are divided into outer and inner cones for both lower and upper hemispheres of the target chamber. The incident angles of outer cones are 55∘ and 49.5∘, and the incident angles of inner cones are 28.5∘ and 35∘, respectively33,34. The sketch of experimental setup and the laser pulse used in the experiment are shown in Fig. 1. The 48 laser beams are injected into a cylindrical gold hohlraum. The diameter and length of the hohlraum are 2200 μm and 4000 μm, respectively. The thickness of the gold hohlraum wall is 30 μm. The diameter of the upper LEH is 1400 μm and the lower end is open. Both ends of the hohlraum are covered by 500 nm thick polyimide films. The hohlraum is filled with C5H12 gas at a density of 1.3 mg/cc. The beams A1 and A2 in the outer cone of 55∘ at lower hemisphere are used as interaction beams with a power of 1 TW per beam, and other 46 beams are used as heating beams with a power of 0.72 TW per beam. The diameters of beam spots at LEHs are 400 μm for the interaction beams A1 and A2 to generate a nominal laser intensity of 1.4 × 1015 W/cm2 and 500 μm for the heating beams, respectively. The wavelength of all laser beams is 0.351 μm, so there is no wavelength separation between the inner and outer cones. The experiments case I and case II are carried out for comparison. The laser configuration and polarization arrangement of the 24 beams of lower hemisphere are shown in Fig. 2a for case I and in Fig. 2c for case II, respectively. Backscattered light of A1 and A2 is diagnosed by FABS as shown in Fig. 1. The main experimental results are shown in Table 1 and Fig. 3. The diagnostic uncertainty of the backscattered light energy is  ± 18%.

The CBET experiments employ a shortened cylindrical gold hohlraum. The thickness of hohlraum wall is 30 μm. The hohlraum is filled with C5H12 gas at a density of 0.8 mg/cc and driven by 24 laser beams from lower hemisphere. To ensure that the laser beams in outer cones heat the hohlraum wall and the laser beams in inner cones pass through the hohlraum, the diameter and length of the hohlraum is set as 2600 μm and 1400 μm, respectively. The diameter of lower LEH is 1400 μm and the upper end is open. The diameter of laser beams at LEH plane is 500 μm. The laser powers of the inner and outer cones are shown in Fig. 6a. To enable energy transfer from outer cones to inner cones, the wavelength separation was \({\lambda }_{in}-{\lambda }_{out}=1.67\,{{\text{\AA}}}\) with λout = 0.351 μm for the outer cones. The beam B1 at 28.5∘ from inner cone is used for CBET diagnostics. After passing through hohlraum, it is directed to a gold witness plate, and SRFD46,47 is used to measure the x-ray radiation emitted from the plate. The measurement uncertainty of SRFD is  ± 10%. The backscattered light of B1 is diagnosed by FABS.

Numerical simulations

We performed the hydrodynamic and CBET simulations for each case two times. In the first round, the designed laser power or the experimental laser power with reduction of backscattering are used as the input to hydrodynamic simulations to obtain evolution of hohlraum plasma parameters. With the simulated hohlraum plasma conditions, we perform the CBET simulation by using the 3-dimensional ray-tracing CBET postprocessing code to obtain the transferred laser power for every laser beam. In the second round, the simulated laser power after CBET is used as the input to hydrodynamic simulation to obtain the hohlraum plasma conditions again. The final laser power after CBET is obtained by using the new hohlraum plasma conditions. And the simulated result of backscattering is obtained by using this final laser power.

The 2D hydrodynamic simulation is performed by the LARED-Integration code41,52. LARED-Integration is a 2D Lagrangian radiation hydrodynamics code for laser target coupling, hohlraum physics and ICF studies42,43,53,54. It has various modules for fluid, laser propagation and absorption, radiative transport, electron heat conduction, nonlocal equilibrium atomic processes, etc. In the LARED-Integration code, the sophisticated opacity and equation of state databases are used for the plasmas in the local thermodynamic equilibrium (LTE) state. The non-LTE state plasmas are described the ideal gas equation of state and the relativistic Hartree-Fock-Slater self-consistent average atom model OPINCH55, which involves the processes of free-free, free-bound, and bound-bound transitions. The electron heat flux is calcluated by a flux-limited model with a flux limiter of fe = 0.1 based on the Spitzer-Härm electron conduction.

The CBET process is calculated by a 3-dimensional ray-tracing postprocessing code. In this code, each laser beam is divided into hundreds of rays. For an arbitrary beam m, zm is the unit vector of its propagation direction, and xm, ym are the unit vectors of two polarization components. The spatial change of vector potential A for each laser ray is obtained by solving the equation,

$$\left(\frac{\partial }{\partial {z}_{m}}+{\kappa }_{m}^{i}\right){{{{\bf{A}}}}}_{m}^{i}={\sum}_{m\ne n}\frac{i\Delta {k}_{mn}^{2}{e}^{2}}{8{k}_{m}{m}_{e}^{2}{c}^{4}}{K}_{mn}^{ij}{{{{\bf{A}}}}}_{m}^{i}\cdot {{{{\bf{A}}}}}_{n}^{j*}{{{{\bf{P}}}}}_{m}\left({{{{\bf{A}}}}}_{n}^{j}\right),$$

where the superscripts i, j denote the laser ray in different beams, the beat wavenumber Δkmn = ∣km − kn∣ and km is the wavevector of beam m, \({\kappa }_{m}^{i}\) is the spatial rate of inverse-bremsstrahlung absorption, K = χe(1 + χi)/(1 + χe + χi) is the plasma response function of electron susceptibility χe and ion susceptibility χi which are respectively the same as those used in refs. 37,38,39, and \({{{{\bf{P}}}}}_{m}\left({{{{\bf{A}}}}}_{n}^{j}\right)=\left({{{{\bf{x}}}}}_{m}\cdot {{{{\bf{A}}}}}_{n}^{j}\right){{{{\bf{x}}}}}_{m}+\left({{{{\bf{y}}}}}_{m}\cdot {{{{\bf{A}}}}}_{n}^{j}\right){{{{\bf{y}}}}}_{m}\) denotes the projection of \({{{{\bf{A}}}}}_{n}^{j}\) on the polarization plane of beam m. As usual, me, e and c are the electron mass, electron charge and light speed in vacuum, respectively. Based on this model, the polarization arrangement can be considered for all beams. Furthermore, the energy transfer between each pair of two beams and the polarization change of each beam due to the energy transfer and phase delay39 can also be calculated.

The backscattering is simulated by HLIP code44,45 which is also a 3-dimensional ray-tracing code to calculate convective amplification of SBS and SRS by dividing the laser beam into hundreds of rays. For the ith ray of beam m, the intensity of incident light \({I}_{m}^{i}\) is calculated by solving the equation,

$$\frac{\partial {I}_{m}^{i}}{\partial {z}_{m}}+2{\kappa }_{m}^{i}{I}_{m}^{i}=-{\omega }_{m}{I}_{m}^{i} {\sum}_{s=R,B}\int_{{\omega }_{s}}\left({\Gamma }_{sm}^{i}{Z}_{sm}^{i}+{T}_{sm}^{i}\right)\frac{d{\omega }_{s}}{{\omega }_{s}},$$

where \({Z}_{sm}^{i}\) denotes the spectral density of SRS or SBS backscattered light with the intensity defined as \({I}_{sm}^{i}={\int}_{{\omega }_{s}}{Z}_{sm}^{i}d{\omega }_{s},(s=R,B)\). Here, the subscript R means SRS and B means SBS, respectively. The \({Z}_{sm}^{i}\) can be obtained by solving the equations,

$$-\frac{\partial {Z}_{sm}^{i}}{\partial {z}_{m}}+2{\kappa }_{sm}^{i}{Z}_{sm}^{i}={I}_{m}^{i}{\Gamma }_{sm}^{i}{Z}_{sm}^{i}+{I}_{m}^{i}{T}_{sm}^{i}+{\Sigma }_{sm}^{i},$$

where \({\kappa }_{sm}^{i}\) denotes the the spatial rate of inverse-bremsstrahlung absorption of the backscattered light of SRS or SBS. \({\Sigma }_{sm}^{i}\) and \({I}_{m}^{i}{T}_{sm}^{i}\) are the noise sources of backscattering from bremsstrahlung and Thomson scattering17,44, respectively. The coupling coefficient

$${\Gamma }_{sm}^{i}=\frac{2\pi ({k}_{m}+{k}_{s})^{2}{e}^{2}}{{k}_{s}{k}_{m}{\omega }_{m}{m}_{e}^{2}{c}^{4}}{{{\rm{Im}}}}\left({K}_{ms}^{i}\right),$$

where, ωm and ωs are the frequencies of incident light and backscattered light respectively, km and ks are the wavenumbers of incident light and backscattered light, respectively. \({K}_{ms}^{i}\) still denotes the plasma response function37,38,39. With this model of HLIP code, the effect of pump depletion, inverse bremsstrahlung absorption, Landau damping, and kinetic growth for different modes of backscattered light can be calculated based on the plasma parameter conditions obtained by the hydrodynamic simulations.