Inertial confinement fusion (ICF)1–3 has attracted significant attention in recent years as a potential method for generating fusion energy. In ICF, a fuel target is compressed to extremely high density and temperature by high-energy lasers or particle beams to trigger the fusion reaction. To achieve better ICF implosion performance, it is necessary to improve the efficiency of coupling between laser energy and target and to control hydrodynamic instabilities,4–6 primarily Rayleigh–Taylor instability (RTI).7,8 RTI occurs on the outer or inner surface of the target during the acceleration or deceleration phase of the implosion in ICF. When the injected energy is deposited on the target, the surface shell absorbs this energy and converts it into a high-temperature and low-density plasma that is ejected outward at high velocity. This ablation process produces a high-pressure area, which drives the fuel to accelerate and compress inward. At the acceleration interface, RTI is triggered by the presence of different gradients of density and pressure. Meanwhile, perturbations at internal interfaces grow, coupled with ablation surface perturbations, leading to material mixing, which can severely damage the integrity of the target shell and affect the compression of the fuel and the efficiency of nuclear fusion reactions.9

The linear theory of single-mode classical RTI shows that an initial sinusoidal perturbation with wavelength λ_p grows exponentially with growth rate ${\gamma }_{\text{CRTI}}=\sqrt{A_T{k}_{p}g}$, where g is the acceleration, k_p = 2π/λ_p is the perturbation wavenumber, and A_T = (ρ_h − ρ_l)/(ρ_h + ρ_l) is the Atwood number. Here, ρ_h and ρ_l are the densities of the heavy and light fluids. With increasing mode amplitude, the perturbed interface evolves into an asymmetric “bubble” and “spike,” and the exponential growth of the fundamental mode saturates. According to the classical theory, when the mode amplitude reaches about 0.1λ_p, the phase of linear growth comes to a halt and the phase of nonlinear growth begins.10,11 It is well known that the ablation process reduces the linear growth rate of RTI, especially when the perturbation wavelength is so short that it approaches the cutoff wavelength. In addition, ablation can significantly reduce the values of the second- and third-order mode coupling coefficients, thereby slowing down the generation of second and third harmonics,12,13 which makes the saturation amplitude of the ablative RTI (ARTI) significantly greater than 0.1λ_p. The linear growth rate of ARTI can be approximated by the Takabe formula14,15${\gamma }_{\text{ARTI}}=\alpha \sqrt{k_{p}g}-\beta k_p{V}_a$, where $V_a=\dot{m}/{\rho }_a$ is the ablation velocity, $\dot{m}$is the mass ablation rate, ρ_a is the density of unablated shell near the ablation front, and α and β depend on the flow parameters. It should be noted that the Takabe formula is applicable in the case of a sharp ablation front16 (k_pL_m << 1, where L_m is the characteristic length of the minimum density gradient near the ablation front), and there are large deviations from this formula when the perturbation wavelength is short and the characteristic width of the ablation front L₀ is large.17 For this reason, Lindl proposed an improved formula18,19$\gamma =\sqrt{k_{p}g/(1+k_p{L}_m)}-\beta k_p{V}_a$ that includes the finite thickness of the ablation front. More complex and accurate self-consistent stability theories20–25 have been developed for the limits of large or small k_pL₀ and large or small Froude number $Fr=V_a^2/g{L}_0$.

The dispersion relation for a laser in a plasma is $c^2{k}^2={\omega }^2-{\omega }_{pe}^2$, where ${\omega }_{pe}={(4\pi n_e{e}^2/m_e)}^{1/2}$ is the plasma frequency. At the location where the plasma frequency equals the laser frequency, the laser is reflected by the plasma, and the electron density at this point is called the critical density n_c. Generally, n_c = [1.115/λ(nm)²]cos θ × 10²⁷cm⁻³,where θ is the incidence angle of the laser.26 The electromagnetic wave power P is depleted by the inverse bremsstrahlung process,27 and the power loss is dP/dt = −ν_ib(t)P, where ν_ib = (n_e/n_i)ν_ei and ν_ei is the frequency of electron–ion collisions. For direct-drive ICF, the typical laser wavelength is 351 nm, and the efficiency of coupling of the laser energy to the kinetic energy of the implosion fluid is about 6%.28–30 For indirect-drive ICF, lasers are incident into a Hohlraum composed of high-Z material, and the laser energy is converted to an X-ray radiation field with a particular energy spectral distribution.31 The energy absorbed by the target ultimately needs to be converted into kinetic energy of the implosion fluid, with the remaining energy consumed to maintain the isothermality of the corona plasma and to ablate cold material. The efficiency of conversion to kinetic energy of the implosion fluid depends on the laser wavelength.

It is well known that in the case of direct-drive ICF, hydrodynamic instability is usually inevitable, owing to the crossing and coupling of the driving lasers. Additionally, the validity of ARTI growth theory at short laser wavelengths has not been verified. In this paper, to better understand and suppress hydrodynamic instability in ICF experiments and investigate how changes in laser wavelength affect ARTI growth, we use numerical simulations to study the effect of laser wavelength on the growth of ARTI. We adjust the intensity to keep the kinetic energy of the implosion fluid constant and thus obtain the relative coupling efficiency under an intensity benchmark. We then investigate laser action on a target for different laser wavelengths and the corresponding intensities. The simulation results show that the ARTI growth rate has its minimum in the extreme ultraviolet band, which represents a deviation from the theoretical behavior of the ARTI growth rate. Finally, the effects of the intensity benchmark and the perturbation wavenumber are investigated.

The remainder of the paper is organized as follows. In Sec. II, the simulation methods and configurations are introduced. In Sec. III, the effects of varying laser wavelength on the growth of ARTI and laser coupling efficiency are presented and discussed. In Sec. IV, cases with different laser intensity benchmarks are presented and discussed. In Sec. V, cases with different perturbation wavenumbers are presented and discussed. Finally, the paper concludes with a summary in Sec. VI.

For the simulations, we use the radiation magnetohydrodynamics code FLASH,32,33 which can handle plasmas with multiple temperatures to simulate laser-driven high-energy-density physics (HEDP) experiments. The relevant equations describing the evolution of a nonmagnetic three-temperature (3T) plasma are as follows:$$\begin{array}{c}\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathit{u})=0,\\ \frac{\partial }{\partial t}(\rho \mathit{u})+\nabla \cdot (\rho \mathit{u}\mathit{u})+\nabla P_{\text{tot}}=0,\\ \frac{\partial }{\partial t}(\rho e_{\text{ion}})+\nabla \cdot (\rho e_{\text{ion}}\mathit{u})+P_{\text{ion}}\nabla \cdot \mathit{u}=\rho \frac{c_{v,\text{ele}}}{{\tau }_{ei}}(T_{\text{ele}}-T_{\text{ion}}),\\ \begin{array}{cc}\hfill \frac{\partial }{\partial t}(\rho e_{\text{ele}})+\nabla \cdot (\rho e_{\text{ele}}\mathit{u})+P_{\text{ele}}\nabla \cdot \mathit{u}& =\rho \frac{c_{v,\text{ele}}}{{\tau }_{ei}}(T_{\text{ion}}-T_{\text{ele}})-\nabla \cdot {\mathit{q}}_{\text{ele}}\hfill \\ \hfill & +Q_{\text{abs}}-Q_{\text{emis}}+Q_{\text{las}},\hfill \end{array}\\ \frac{\partial }{\partial t}(\rho e_{\text{rad}})+\nabla \cdot (\rho e_{\text{rad}}\mathit{u})+P_{\text{rad}}\nabla \cdot \mathit{u}=\nabla \cdot {\mathit{q}}_{\text{rad}}-Q_{\text{abs}}+Q_{\text{emis}}.\end{array}$$Here, ρ is the fluid density, u is the fluid velocity, P_tot= P_ele+ P_ion+ P_rad is the total pressure, τ_ei is the electron–ion energy relaxation time, T_ele and T_ion are the temperatures of the electrons and ions, e_ion, e_ele, and e_rad are the energies of the ions, electrons, and radiation field, c_v,ele is the electron heat capacity, q_ele and q_rad are the heat conduction fluxes of the electrons and the radiation field, Q_las is the laser energy deposition, Q_abs = σ(r, λ, t)I(r, λ, t) is the absorption of radiation by electrons, where σ(r, λ, t) is the opacity of the material and I(r, λ, t) is the radiation intensity, and Q_emis = 4πσ(r, λ, t)B(T, λ) is the emission of radiation by electrons, where B(T, λ) is the Planck function. The pressure, density, temperature, and internal energy are related by the equation of state.

The control equations are solved on a uniform grid, using the fifth-order weighted essentially non-oscillatory (WENO) method34 in an unsplit hydrodynamics solver.35 The laser energy deposition is calculated using the EnergyDeposition unit of FLASH, the 3T hydrodynamics update is performed using the Hydro unit, the 3T equation of state is calculated using the Eos unit, ion/electron equilibration is implemented using the Heatexchange unit, the implicit diffusion solver is implemented using the Diffuse unit, the electron thermal conductivity is calculated using the Conductivity unit, the vanLeer slope limiter36 is used to deal with shock waves, and a second-order accurate HLLC Riemann solver37 is used to calculate time-advanced fluxes. A Courant–Friedrichs–Lewy (CFL)38 number of 0.4 is adopted to ensure numerical stability.

To obtain a steady-state flow field, we use a laser with wavelength 351 nm and intensity39,40 1 × 10¹⁴ W/cm² to simulate the laser-driven ablation of a one-dimensional planar CH target of thickness 200 μm and density 1 g/cm³. After a few nanoseconds, the ablation surface reaches uniform acceleration motion, meaning that the state is steady.41 We then extend the steady-state flow field into two dimensions and introduce perturbations. Figure 1(a) shows the initial profiles of density, temperature, and velocity along the z axis. Figure 1(b) shows the simulation setup for the initial density profile. The upper part is the cold and dense unablated material, while the lower part is the hotter but less dense ablated plasma. The size of the simulation domain is 25.6 × 300 μm², with a grid size of 0.1 μm,42,43 and periodic boundary conditions are set at the left and right boundaries. An initial single-mode perturbation with amplitude η = η₀ cos(k_px), where η₀ = 0.1 μm and k_p = 2π/12.8 μm⁻¹, is introduced at the ablation front, which is realized by shifting the one-dimensional fluid when it is extended to two dimensions. A laser with a varying wavelength but a constant intensity and a constant duration of 4 ns is vertically incident from the bottom.

Studies have revealed that the coupling efficiency between laser and target during the implosion acceleration phase depends on the laser wavelength. To control variables, we take the initial laser parameters as the benchmark (with an intensity of 1 × 10¹⁴ W/cm² and a wavelength of 351 nm) and then change the laser wavelength and adjust the intensity so that the state (implosion velocity and position of the ablation surface) of the target is consistent after 4 ns of laser action, as shown in Fig. 2. The ratio of the laser intensity used in the simulation to the benchmark is taken as the relative coupling efficiency between the input laser and the kinetic energy of the implosion fluid, as shown in Fig. 3. Here, we compare the results obtained by this method (red circles) with those from simulations and three analytic models (models A, B, and C) provided in Ref. 44. When λ ≥ 200 nm, our results match well with the simulation results from Ref. 44 (blue crosses), and when λ < 200 nm, they are in closer agreement with those of model B (green downward-pointing triangles). We believe that since the simulation results in Ref. 44 were fitted using λ = 193, 248, 351, and 527 nm, they are more accurate within this wavelength range than those of the model. For λ < 193 nm, the assumption used in model B that the laser is absorbed by inverse bremsstrahlung before the critical surface is appropriate. According to Fig. 3, the shorter the laser wavelength, the higher is the relative coupling efficiency. Here, the laser energy is acting directly on the target. When λ = 10 nm, the electromagnetic wave begins to enter the X-ray band, and the relative coupling efficiency at this wavelength is 3.5 times that at λ = 526 nm.

On the basis of the relative coupling efficiency, we investigate laser action on a target with initial perturbation amplitude η = η₀ cos(k_px), where η₀ = 0.1 μm and k_p = 2π/12.8 μm⁻¹, for different laser wavelengths and corresponding intensities. The ARTI perturbations near the ablation surface after 4 ns of laser action are shown in Fig. 4, and the corresponding average ARTI growth rates γ = ln(η/η₀)/t within 4 ns are shown in Fig. 5. We find that the ARTI growth rates are larger for both longer and shorter wavelengths. However, as the wavelength moves from 150 nm toward the extreme ultraviolet band, the ARTI growth rate decreases rapidly, reaching a minimum value of γ = 0.31 ns⁻¹ near λ = 65 nm. However, as the wavelength moves continuously toward the X-ray band, the ARTI growth rate increases rapidly.

To explore the causes of the variation of the average ARTI growth rate with laser wavelength, we analyze the variations of the main hydrodynamic parameters (g, L_m, V_a, conduction layer thickness L_c, and A_T) with laser action time at five representative wavelengths (351, 100, 65, 40, and 30 nm), as shown in Figs. 6(a)–6(e), and then we calculate the variation of the theoretical ARTI growth rate with laser action time according to the Lindl formula,18,19 as shown in Fig. 6(f). In Fig. 7, we compare the variations with laser action time of the simulated perturbation amplitude for different laser frequencies (symbols) and the perturbation amplitude calculated from the theoretical ARTI growth rate (dashed line). We find that the shorter the laser wavelength, the smaller is the theoretical ARTI growth rate, which is mainly due to the fact that V_a increases as the laser wavelength becomes shorter, while the influence of other hydrodynamic parameters is small. For λ = 351 and 100 nm, the simulated perturbation amplitude is close to the theoretical results, but for λ = 65 nm, it is smaller than these, and in particular, for λ = 40 and 30 nm, the theoretical ARTI hardly grows, which represents a large deviation from the simulated perturbation amplitude.

After further analysis of the fluid state, we deem there to be two main causes of the deviation. First, when λ < 50 nm, the maximum value of the plasma density is less than the critical density, and under this condition, the laser energy can reach the ablation surface in the form of electromagnetic waves instead of being absorbed by the plasma before the critical surface, as a consequence of which the electromagnetic wave is deflected in the perturbation region of plasma density, as shown in Fig. 8. Figure 8(a) shows how the direction of the laser changes in the perturbation region of plasma density, and Fig. 8(b) shows the distribution of laser energy deposition at λ = 30 nm. For comparison, Fig. 8(c) shows the distribution of laser energy deposition at λ = 65 nm. It can be seen that owing to the deflection of the laser, the energy flow distribution for ablation on the ablation surface becomes uneven in the transverse direction, and as the laser wavelength continues to decrease, the proportion of laser energy reaching the ablation surface in the form of electromagnetic waves increases, and so the energy flow distribution is more uneven, making the ARTI perturbation amplitude larger.

The second cause of the deviation is that for longer wavelengths (>100 nm), the laser energy is absorbed before the critical surface, and the energy flow then has to pass through a thicker electron conduction layer to reach the ablation surface, as shown in Fig. 6(d). Because of the existence of ARTI, there is a perturbation of the temperature distribution in the electron conduction layer, as shown in Fig. 9, where A and B are two adjacent positions in the transverse direction, with A having a higher temperature than B. At this time, an energy flow is generated from A to B, with energy flow density q = −κ∂T_e/∂x, where $\kappa \propto T_e^{5/2}$ is the coefficient of thermal conductivity. When λ = 65 nm, the critical density is close to the plasma density at the ablation front. At this time, the electron conduction layer is very thin, and so the transverse energy flow generated as the energy flow reaches the ablation surface is much smaller than that in the long-wavelength case. Therefore, for λ ≤ 65 nm, the transverse distribution of energy flow reaching the ablation surface effectively becomes uniform if the laser deflection is not considered. This effect is not taken into account in the ARTI growth rate formula, and so it is necessary to add a correction term related to L_c (or laser wavelength), which makes the formula more applicable to short-wavelength situations.

To study the effects of different values of the intensity benchmark I_b, we consider intensities of 2 × 10¹⁴ and 4 × 10¹⁴ W/cm² with a wavelength of 351 nm and simulate laser-driven ablation of a one-dimensional target. We then extend the steady-state flow field into two dimensions and introduce perturbations again. The size of the simulation domain is now set to 25.6 × 600 μm², because of the increased implosion velocity. We find that the relative coupling efficiency between lasers with different wavelengths and the kinetic energy of the implosion fluid remains the same for different intensity benchmarks. The ARTI average growth rate γ is shown in Fig. 10 for an initial single-mode perturbation with k_p = 2π/12.8 μm⁻¹. The growth rate reaches a minimum near λ = 55 nm for I_b = 2 × 10¹⁴ W/cm² and near λ = 40 nm for I_b = 4 × 10¹⁴ W/cm², which are shifts of about 10 nm and about 25 nm, respectively, toward shorter wavelengths. This is because the ablation velocity and pressure increase with increasing intensity benchmark, which results in an increase of plasma density at the ablation surface, reducing the laser wavelength needed for a noticeable deflection effect. On the other hand, the increase in ablation velocity enhances the stability of ablation, resulting in a minimum growth rate lower than that at I_b = 1 × 10¹⁴ W/cm².

In ICF, many perturbation modes grow simultaneously. During the linear growth phase, the Lindl formula can predict the growth of perturbations well under ablation conditions, and initial perturbations with mode numbers l = 140–760 eventually enter the nonlinear growth phase,45 where l is the Legendre polynomial mode of the surface perturbation amplitude. For a typical ICF spherical target with a radius of 1 mm, the corresponding perturbation wavelength range is 8–45 μm. Therefore, we initially select a wavenumber k_p = 2π/12.8 μm⁻¹, and we study cases in which the single-mode perturbation wavenumber k_p = 2π/25.6 μm⁻¹ or 2π/51.2 μm⁻¹ for I_b = 1 × 10¹⁴ W/cm². The results are shown in Fig. 11. The trend of variation of the ARTI growth rate with laser wavelength is basically consistent with that at k_p = 2π/12.8 μm⁻¹, but the growth rate reaches its minimum near λ = 75 nm for k_p = 2π/25.6 μm⁻¹, which is a shift of about 10 nm toward longer wavelength. For k_p = 2π/51.2 μm⁻¹, the ARTI growth rate reaches its minimum near λ = 95 nm. This is because the smaller the perturbation wavenumber, the longer is the perturbation region of plasma density, as shown in Fig. 12. This causes a laser with a given wavelength to start deflecting at a position farther away from the ablation surface as the perturbation wavenumber decreases, and so the laser wavelength needed for a noticeable deflection effect becomes longer.

The effect of laser wavelength on the growth of ARTI has been studied through numerical simulations. In the plasma acceleration phase, shorter wavelengths lead to more efficient coupling between the laser and the kinetic energy of the implosion fluid. Under the condition that the laser energy coupled to the implosion fluid is constant, the ARTI growth rate decreases as the wavelength moves toward the extreme ultraviolet band, reaching a minimum value of γ = 0.31 ns⁻¹ near λ = 65 nm. As the wavelength continuously moves toward the X-ray band, the ARTI growth rate increases rapidly. This is due to three causes. First, the ablation velocity V_a becomes larger as the laser wavelength decreases, which enhances the stability of ablation. Second, when the maximum value of the plasma density is less than the critical density, laser energy can reach the ablation surface in the form of electromagnetic waves, which are deflected in the perturbation region of plasma density, resulting in an energy flow distribution for ablation on the ablation surface that is uneven in the transverse direction. Under this condition, the theoretical ARTI growth rate deviates significantly from the simulated results. Third, for longer wavelengths, the energy flow has to pass through a thicker electron conduction layer to reach the ablation surface, and the temperature distribution in the electron conduction layer is perturbed by the presence of ARTI, which leads to the generation of a transverse energy flow and thus to an uneven transverse distribution of energy flow reaching the ablation surface. It is therefore necessary to modify the ARTI growth formula.

As the laser intensity benchmark increases, the position of the minimum ARTI growth rate shifts toward shorter wavelengths. This is because the ablation velocity and pressure increase with increasing intensity benchmark, which results in an increase in plasma density at the ablation surface, as a consequence of which the laser wavelength needed for a noticeable deflection effect becomes shorter.

As the initial perturbation wavenumber decreases, the position of the minimum ARTI growth rate shifts toward longer wavelengths. This is because the smaller the perturbation wavenumber, the longer is the perturbation region of plasma density, and a laser of given wavelength starts to be deflected at a position farther away from the ablation surface.

Thus, numerical simulation and theoretical analysis have indicated that extreme ultraviolet lasers of specific wavelengths have higher coupling efficiency and can better control the growth of instability, thereby significantly reducing the laser energy required for fusion ignition and achieving higher fusion energy gain. At present, the main methods to generate high-power extreme ultraviolet light sources are by multiplying the frequency of high-power lasers46 or by generating high-order harmonics through laser interaction with solid materials.47,48 Therefore, the use of extreme ultraviolet lasers to achieve ICF ignition is an important topic worthy of further investigation in the future.

Acknowledgment. This work is supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 12074399, 12204500, and 12004403), the Key Projects of the Intergovernmental International Scientific and Technological Innovation Cooperation (Grant No. 2021YFE0116700), the Shanghai Natural Science Foundation (Grant No. 20ZR1464400), and the Shanghai Sailing Program (Grant No. 22YF1455300).