With the milestone of inertial confinement fusion (ICF) ignition1–3 having been achieved at the National Ignition Facility (NIF), the next goal of higher energy gain is now being pursued by the ICF community. Direct-drive schemes, in which high-intensity lasers directly deposit energy in an ICF target, generally have high laser-to-target energy coupling efficiency and are considered as candidates for high-gain fusion approaches.4 Direct laser interaction with the target is not only a key process in the conventional central-hot-spot direct-drive scheme, but is also utilized in one or multiple phases of novel ignition schemes such as shock ignition,5 hybrid ignition,6 and double-cone ignition.7

One of the critical challenges faced by direct-drive ICF is the harmful impact of laser–plasma instabilities (LPIs), including stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), cross-beam energy transfer (CBET), and two-plasmon decay (TPD) instability, occurring in the corona plasmas ablated from the target shell. As a laser propagates through a direct-drive corona plasma, it experiences a broad range of electron density from vacuum up to the critical density n_cr beyond which a laser is not able to propagate. In this scenario, the areas of quarter-critical density n_cr/4, where TPD and absolute SRS can be parametrically stimulated, are usually located in a less-collisional plasma consisting of low-Z ablator materials and are therefore of critical concern. The LPI processes near n_cr/4 are of less concern in an indirect-drive Hohlraum, since the n_cr/4 areas are usually located in a high-Z gold plasma where the heavy collisional damping inhibits the growth of TPD and SRS. LPIs can not only cause direct energy loss through the processes involving scattered lights, i.e., SRS, SBS, and CBET, but also lead to a risk of fuel preheating through the energetic (hot) electrons produced via TPD and SRS, which involve electron plasma waves (EPWs) that are able to accelerate electrons.

Through TPD, a laser parametrically decays into a pair of so-called daughter EPWs. TPD is considered as a critical LPI impacting the implosion performance in direct drive ICF, owing to its low threshold8–10 and hot-electron generation.11,12 TPD has been identified as the primary source of hot electrons that can preheat the fuel4,11 and thus compromise implosion performance on the OMEGA direct-drive facility.

Although modern ignition facilities generally use multiple overlapping laser beams, much of the knowledge on TPD was first established in simplified single-beam regimes.8,9,12–18 The threshold for TPD growth in an inhomogeneous plasma with a linear density profile in a normal-incidence regime [i.e., one where the laser propagation direction is along the electron density (n_e) gradient ∇n_e] was obtained9 asη≡LμλμI14/(82TkeV)>1,(1)where L_μ is the electron density scale length L_n at n_cr/4 in units of micrometers, λ_μ is the laser wavelength in vacuum (λ₀) in units of micrometers, I₁₄ is the laser intensity I₀ in units of 10¹⁴ W/cm², and T_keV is the electron temperature T_e in units of keV. In Refs. 17 and 18, two different types of modes, absolute and convective, were identified via fluid simulations on the basis of their distinct growth behaviors in a linear system. Absolute modes grow exponentially with time, whereas convective modes have limited factors of amplification on the initial seeds. However, simulations17 have shown the convective modes to be energetically important in, owing to nonlinearity involving ion dynamics,12,15–17 and therefore both absolute and convective modes saturate, forming quasi-static broad EPW spectra up to the Landau cutoff in k space. Broad spectra of the 3ω₀/2 emission associated with TPD EPWs, where ω₀ is the laser frequency, have been consistently observed in OMEGA experiments.19

It is well recognized that TPD can be collectively driven by multiple overlapping lasers. The total energy in hot electrons has been shown experimentally on OMEGA to scale with the overlapped intensity.20 The threshold condition of Eq. (1), which has been widely verified by fluid simulations,18 also works empirically, giving a good prediction of the onset of TPD in OMEGA experiments if the overlapped intensity is used to calculate η.19 These results show that TPD is highly unlikely to be independently driven by each individual laser beam. Experiments have been carried out to investigate the growth of the shared daughter plasma waves (often referred to as the common waves) contributed by multiple laser beams.21,22 It has been found that laser beams can drive the common waves in the region of wavenumbers bisecting the beams and that the convective gain is proportional to the overlapped laser intensity.21 Moreover, the hot-electron fraction has been found to increase exponentially with the overlapped laser intensity in experiments with various beam configurations.22 Three-dimensional (3D) simulations have further shown that the threshold for the absolute TPD modes driven by multiple beams is significantly lower than the expected single-beam value.23

An important factor that may contribute to the low threshold found in Ref. 23 comes from the oblique incidence angles. The single-beam threshold intensity of TPD under moderate incidence angles has been found to be lowered by a factor of ∼cos²ϕ,24 where ϕ is the angle between the local laser wave vector k_loc and ∇n_e near n_cr/4. Large incidence angles, however, lower the absolute TPD threshold even more dramatically.25,26 It should be noted here that in this paper, the incident angle θ is taken to be the angle between k₀ and ∇n_e, where k₀ is the laser wave vector in vacuum with magnitude k₀ ≡ 2π/λ₀ = ω₀/c and c is the speed of light in vacuum. The regime in which θ is close to 60° or even larger is considered to be the large-angle-incidence regime. On the basis of theoretical modeling and fluid simulations using the LTS fluid code,18 which solves the linear TPD equations8 and has been shown to give a good description of the linear growth of TPD modes,18,25 Lian et al.25 proposed a threshold for the absolute TPD growth in the large-angle-incidence regime with θ = 60°:ξ≡Lμ4/3λμ2/3I14/(3TkeV)>1,(2)where ξ is the threshold parameter in the large-angle-incidence regime and TPD can be driven when ξ > 1, which has also been found to be consistent with particle-in-cell (PIC) simulations.25 This large-angle-incidence TPD threshold has been shown to be two orders of magnitude lower than that for normal incidence with ICF-relevant parameters.25 Moreover, simulations of TPD induced by multiple beams all with θ ≈ 60° also exhibited low threshold and common-wave processes that largely depend on the lasers’ polarization.26

The very low TPD threshold in the large-angle-incidence regime draws attention to the case of TPD in a complicated laser–plasma system where virtually all incidence angles are present and a portion of the plasma can be simultaneously irradiated by both small- and large-angle-incidence beams. Whether the large-angle-incidence TPD that can be driven at low laser intensities can cause destabilizing effects in the whole system is an open question.

In this paper, we study TPD in a simpler but typical “N-L” two-beam system composed of a normal-incidence beam (Beam-N) and a large-angle-incidence beam (Beam-L) via numerical simulations. A schematic of an N-L system in a direct-drive scheme is illustrated in Fig. 1(a). Although each beam in a direct-drive scheme is generally pointed at the target center, the off-axis part of a laser with weaker intensity than the peak value will graze the edge of the corona plasma surrounding the target. Therefore, the zone inside the box in Fig. 1(a) can be considered to be irradiated by a higher-intensity Beam-N and a lower-intensity Beam-L. Such N-L systems typically occur in many direct-drive schemes such as conventional central-hot-spot ignition,4 shock ignition,5,27 and double-cone ignition.7

The remainder of this paper is organized as follows. The simulation setup is described in Sec. II. The development of TPD in an N-L configuration is investigated in Sec. III, and the generation of hot electrons is discussed in Sec. IV. A summary is presented in Sec. V.

A series of PIC simulations are performed using the full PIC code OSIRIS28 to investigate the LPI processes in the boxed zone in Fig. 1(a). A schematic of a typical simulation box is presented in Fig. 1(b). The parameters chosen are relevant to OMEGA and SG direct-drive experiments. The plasma has a linear electron density profile n_e(x) = (0.20 + 0.25x/L_n)n_cr and a uniform electron temperature T_e = 3 keV, where the electron density scale length is set as L_n = 100 μm in all of the simulations in this paper. No background plasma flow is set up. In contrast to other types of LPIs such as SBS and CBET, which are extremely sensitive to the ambient plasma flow, the plasma flow has been found to have a subdominant effect on TPD, resulting in a 10%–15% increase in the absolute threshold for a Mach 1 flow according to the theory of multiple-beam TPD.23 The ions are composed of fully ionized CH (plastic) with a number ratio 1:1 and a temperature T_i = 1.5 keV unless otherwise noted. The electron–ion collisions are realized in OSIRIS by a binary collision operator,29,30 while the algorithm for pairing of particles is adapted from those of Nanbu and Yonemura30 and Takizuka and Abe.31 Two laser beams, namely, Beam-N and Beam-L, are launched from the left side with a rise time of 0.2 ps and at incident angles θ = 0° and 58.7°, respectively. Both beams are incident as plane waves. The simulations are set up to mimic a typical area illustrated in Fig. 1(a). The transverse size (20 μm) of the area is much smaller than the diameter of an actual direct-drive Gaussian beam with a large spot size (e.g., 500 μm). Plane waves with periodic boundaries seem to be a reasonable approximation for this scenario. Different combinations of the laser intensities of Beam-N (I_N) and Beam-L (I_L) are explored, and the detailed parameters are listed in Table I. Since Beam-L is abstracted from the off-axis part of an actual laser beam [see Fig. 1(a)], it is physically appropriate to set I_L much weaker than I_N. Both beams are linearly polarized in the x–y plane.

The simulation box size is 34 × 20 μm² with a grid of 3000 × 1820, yielding a grid size Δx = Δy = 0.2c/ω₀. One hundred particles per cell are used for each individual species (i.e., 100 particles for electrons, 100 particles for C, and 100 particles for H per cell). In OSIRIS, the particles are variably weighted on the basis of the initial local charge density in each cell.28 The collision operator30 takes account of variably weighted particles. For pairs of particles with different weights, momentum and energy are not conserved in individual collisions, but they are statistically conserved over a large number of collisions among particles with different weights.30 The simulation domain is periodic in y. Open boundary conditions are applied to the electromagnetic fields in the x direction. The “thermal-bath” boundary conditions, which reflect the particles reaching the boundary and assign them new random velocities following the initial Maxwellian distribution, are applied to the particles in the x direction.

The simulation parameters are chosen such that I_N is well below the threshold, namely, η ≤ 0.71, and so the normal-incidence Beam-N is not intense enough to excite TPD by itself. Beam-L is close to glancing incidence near the turning point approximately located at n_e = 0.27n_cr. The laser fields in case I are shown in Fig. 1(c). In this case, ξ = 13 ≫ 1, indicating that Beam-L is well above the TPD threshold and able to drive absolute growth in the large-angle-incidence regime if shined alone. It should be noted that the “overlapping η” (η_all) calculated by substituting I_all ≡ I_N + I_L into Eq. (1) is also below unity in all of the cases, as listed in Table I. A snapshot of the electron density fluctuations n_p associated with EPWs in case I is shown in Fig. 1(d), and the time and space evolution of ⟨np2⟩y(x,t) is illustrated in Fig. 2(a). Here, the brackets ⟨⋯⟩_y denote averaging over y. It can be seen that EPWs with notable amplitude are driven near n_cr/4 and last throughout the simulation. These EPWs can be attributed to TPD rather than SRS, since virtually no SRS-scattered light appears in the spectrum of the z component of the magnetic fields (B_z), which is associated with the lights. The presence of the large-angle-incidence Beam-L is identified as the key factor enabling TPD growth in case I, since no TPD is observed if we redistribute all the intensity from Beam-L to Beam-N while maintaining the same I_all (see case III of Table I).

Once TPD is able to grow, it can become energetically important and cause significant pump depletion in the N-L system. Figure 2(a) demonstrates the pump-depletion fraction of Beam-N in case I. Roughly 7% of the Beam-N energy is depleted once TPD has been stimulated, which is evidence that Beam-N also participates in the TPD process. Changing the intensity of either Beam-L or Beam-N is found to change the pump depletion fraction and ⟨np2⟩, which can be used to assess the level of TPD. Here, ⟨np2⟩ is the average of np2 over the whole simulation domain. In case II, the intensity of Beam-N is increased to 5 × 10¹⁴ W/cm², corresponding to η = 0.71, which is still lower than the normal threshold, while the intensity of Beam-L is kept the same as in case I. The pump depletion of Beam-N rises to roughly 15%. It should be pointed out that the energy that Beam-N loses can be even higher than the input energy of Beam-L. In cases III–V, different I_L are used while I_N is kept the same as in case I to further demonstrate the sensitive dependence of TPD on the fairly low-intensity Beam-L. Figure 2(c) compares ⟨np2⟩ with different intensities of Beam-N and Beam-L. It can be seen that ⟨np2⟩ saturates to quasi-steady values after a few picoseconds of growth. It can also be seen that raising either I_N or I_L leads to higher TPD levels, and in the absence of Beam-L (case III), TPD is completely turned off. Both the influence of the low-intensity Beam-L and the significant pump depletion of the high-intensity Beam-N are evidence that TPD is collectively driven by both beams.

The electron density perturbation in case I is displayed in k_y–t space to illustrate the evolution of EPWs from the linear growth stage up to the highly nonlinear stage [see Fig. 2(d)]. The narrowband EPW spectra concentrated near k_y ≈ 0.9ω₀/c and 0.07ω₀/c in the early linear stage become much broader in k_y in the highly nonlinear stage. The modes initially growing in the linear stage can be attributed to Beam-L. The dispersion relation for single-beam TPD growth in homogeneous plasmas predicts the well-known fastest-growing hyperbola32 shown by the solid curves in Fig. 3(a), red for Beam-L and blue for Beam-N. The dominant modes lying on top of the red hyperbola also satisfy the wave-vector matching condition illustrated by the arrows:kL=k1+k2,(3)where k₁ and k₂ (green arrows) are the wave vectors of the paired daughter EPWs, and k_L is the local wave vector of Beam-L (black arrow), which propagates along the y direction near its reflection point.

The individual roles of the two beams in developing TPD in the N-L system are further demonstrated by separating their time windows in case VI. In case VI, Beam-L is set to be incident from 0 to 2 ps, while the onset of Beam-N is set at t = 2 ps, so that they barely overlap in time. All the other simulation parameters are kept essentially the same as in case I, except that the ions are fixed in case VI to inhibit nonlinear effects brought about by the ion dynamics. The time evolution of the spectrum of n_p is illustrated in Fig. 3(b). It can be seen that similar to case I in the linear stage [see Fig. 2(d)], the dominant modes in case VI are located at k_y ∼ 0.07ω₀/c and k_y ∼ 0.9ω₀/c, lying on top of the red fastest-growing hyperbola.

We call the modes with k_y ∼ 0.07ω₀/c the small-k_y modes, and those with k_y ∼ 0.9ω₀/c the large-k_y modes. The large-k_y modes are able to grow to a substantial level before the onset of Beam-N, showing that the large-k_y modes can be stimulated by Beam-L alone. During the incidence of Beam-N (>2 ps), the amplitudes of the large-k_y modes are further amplified and then decay after 4 ps. The amplification after 2 ps shows that Beam-N transports energy to the large-k_y modes, while the decay after 4 ps shows that Beam-N by itself cannot maintain the level, which is a feature of a convective amplification.

The evolution of the large-k_y modes in case VI can be divided into two phases: excitation by Beam-L and amplification by Beam-N. The spectra of the EPWs in case VI in k_y–k_x space are depicted in Fig. 3(c) at t = 2 ps and Fig. 3(d) at t = 3 ps to illustrate the TPD matching conditions and the shifting of the dominant modes at different phases. In the excitation phase [see Fig. 3(c)], the spectrum of case VI exhibiting both large-k_y and small-k_y modes is similar to that of case I [Fig. 3(a)]. This demonstrates that Beam-L itself can stimulate the large-k_y and small-k_y modes simultaneously, which satisfy the wave-vector matching condition in Eq. (3) indicated by the solid arrows. Compared with Fig. 3(a), the spectrum in Fig. 3(c) misses a portion near the blue hyperbola, owing to the absence of Beam-N, as marked by the blue circle. The missing portion can be seen in Fig. 3(d), which shows the dominant modes in the amplification phase with Beam-N only. The dominant k_x of both large-k_y and small-k_y modes becomes larger. The shifting of the dominant modes can be attributed to two processes. First, by satisfying its own matching conditions, Beam-N can drive and amplify the paired EPWs (indicated by the green dashed arrows) of the existing EPWs (indicated by the green solid arrows) excited by Beam-L. Second, |k_x| of the existing EPWs increases as they propagate toward lower density, as indicated by the blue flow arrows. The propagation of the EPWs and the convective amplification on the path for the large-k_y and small-k_y modes are further illustrated in Figs. 3(e) and 3(f).

A set of typical small-k_y modes with k_y = 0.07ω₀/c are localized near x = 18 μm immediately below n_cr/4 at t = 2 ps after the absolute growth in the excitation phase driven by Beam-L only, as shown in Fig. 3(e). The amplitude of n_p along the black dashed line is plotted by the black solid line to show the location of the dominant modes at t = 2 ps. Then, instead of continuing to grow locally, the small-k_y modes are found to propagate to different locations and become amplified at certain densities. The presence of three distinct propagation paths in Fig. 3(e) suggests that at t = 2 ps, the small-k_y modes include a range of k_x, leading to different group velocities that follow different propagation paths in t–x space. As the EPWs eventually move to different densities, |k_x| changes, while |k_y| remains the same, since the nonuniformity of density is in the x direction. This difference between k_x and k_y results in the large range of k_x and the narrow range of k_y, forming the narrowband spectra. The upshift of |k_x| can be seen in Fig. 3(d), as marked by the blue flow arrow on the small-k_y modes. The profile of the small-k_y modes at t = 4.5 ps along the red dashed line is plotted by the red solid line in Fig. 3(e), demonstrating the convective amplification of the small-k_y modes by Beam-N.

The convective amplification of a typical large-k_y mode with k_y = 0.9ω₀/c is shown in Fig. 3(f). The mode is convectively amplified slightly from t = 2 ps (black solid line) to t = 2.2 ps (red solid line) as the wave packet moves forward. Compared with the small-k_y modes, the amplification of this mode is much weaker. The difference in the amplification factor between the large-k_y and small-k_y modes can be estimated by the Rosenbluth-like gain presented in Ref. 17:πΛ≈2.15(1−0.00881TkeV−0.0470TkeVk̃y2)η,(4)where the Rosenbluth amplification factor of a mode with different k̃y is f(k̃y)=exp(πΛ), where k̃y≡kyc/ω0 is a dimensionless k_y. Landau damping is absent in Eq. (4). The value f(0.07) = 2.5 gives the same order of magnitude of the amplification factor as that in the PIC simulation of case VI, namely, f_sim(0.07) ≈ 1.8 from Fig. 3(c). According to Eq. (4), the amplification factor with larger k_y is smaller, which is qualitatively consistent with the simulation. However, the gain for the mode with k_y = 0.9ω₀/c obtained by Eq. (4) is much larger than the simulation results. One reason for the discrepancy is that Eq. (4) assumes that the modes are located on the fastest-growing TPD hyperbola, while the mode with k_y = 0.9ω₀/c is far away from the fastest-growing TPD hyperbola of Beam-N as illustrated in Fig. 3(a). Another reason may come from the heavier Landau damping of the EPWs, which is not taken into account in Eq. (4).

Overall, the growth of TPD in the N-L system can be recognized as a “seed-amplification” process in which a daughter EPW is shared between the absolute mode driven by Beam-L and the convective mode driven by Beam-N, as illustrated in Fig. 4. The EPW with k₁, acting as a daughter wave of the absolute mode driven by Beam-L [Fig. 4(a)], is able to also act as a daughter wave of the convective mode driven by Beam-N by satisfying the TPD matching condition [Fig. 4(b)]. In this case, the EPW with k₁ is the shared daughter wave that is commonly driven by both Beam-L and Beam-N, as illustrated in Fig. 4(c). It should be pointed out that k₁ is not located on the fastest-growing hyperbola of Beam-N [see the blue dashed curve in Fig. 4(b)], which results in a lower gain for this mode compared with the most-resonant modes on the fastest-growing hyperbola. However, the EPW with k₁ is energetically dominant, even with lower convective gain, owing to the extraordinarily high seed level caused by the absolute growth driven by Beam-L. The two beams contribute to the TPD growth in different ways, both making TPD energetically important in the N-L system, even after the saturation due to the nonlinear effects, including ion motions.

The EPW spectrum in the nonlinear stage at 4 ps is much broader [Fig. 5(a)] than that in the linear stage [Fig. 3(a)] of case I. The ion dynamics are recognized as a key factor in broadening the EPW spectrum. As shown in Fig. 5(b), case VII with immobile ions and virtually the same other configurations as case I has a much narrower EPW spectrum in k_y than case I. Spectral broadening was also found in simulations of normal-incidence TPD with ion motions in the nonlinear stage.12,17 The ion motions typically have two sources: wave-like ion density fluctuations and large-scale background drift (plasma flow). The ion motions discussed in this paper are essentially the former, while the latter is critical to SBS and CBET but less important for TPD. No evidence of CBET is seen in our simulations, which is consistent with the knowledge that CBET cannot satisfy the three-wave matching condition when the frequencies of the two laser beams are identical and in the absence of a plasma flow. A plasma flow may favor the growth of CBET in this N-L scenario, but this is beyond the scope of the present study. It is well known that ion density fluctuations can be driven by the ponderomotive pressure of the EPWs and are correlated with the level of EPWs.17 In case I, where both beams are shining from the very beginning of the simulation, the ion density fluctuations are gradually driven up and mostly localized in a fairly narrow region near n_cr/4, where the absolute modes are located, as shown in Fig. 5(c).

The ion motions also contribute to TPD saturation and limit the level of EPWs. Figure 5(d) compares the time evolution of ⟨np2⟩, which quantifies the level of EPWs, of case VII with fixed ions and of case I with mobile ions. It can be seen that TPD in case I saturates at a lower level after t = 3 ps when the ion density fluctuations are significant [see Fig. 5(c)]. It has been demonstrated experimentally that the ions can saturate TPD via the Langmuir decay instability,33 in which a primary TPD EPW decays into an ion acoustic wave and a new EPW. By contrast, TPD in case IV saturates later and at a higher level, probably because of the limits on convective amplification18,34 and/or other kinetic saturation mechanisms limiting the EPW amplitudes. It should be pointed out that the inclusion of ion motions is critical for correct modeling of TPD saturation, which largely determines the EPW features and the hot-electron generation in the nonlinear stage.

Hot-electron generation is of great concern, since the hot electrons generated by TPD are recognized as a significant target preheating risk. Usually, it is hot electrons with kinetic energy above 50 keV that are considered the most threatening, and therefore most attention is paid to these in our simulations. Both the forward- and backward-moving instantaneous energy fluxes carried by hot electrons above 50 keV are monitored in the simulations at the right and left boundaries of the simulation box as α_R and α_L, respectively. Here α_R and α_L have been renormalized to the incident laser energy flux including both beams. Then, α₅₀ ≡ α_L + α_R can be recognized as the energy conversion fraction from laser to hot electrons >50 keV.

The time evolution of the αs in case I shown in Fig. 6(a) indicates that substantial laser energy can be transferred to hot electrons, mostly in the nonlinear stage after t = 3 ps. α₅₀ increases sharply at about t = 3 ps and reaches a significant level of ∼7% later in the nonlinear stage, which is correlated with the result that the pump depletion of Beam-N reaches high levels after about 3 ps, as plotted in Fig. 2(a). This correlation is evidence that the majority of the hot-electron energy comes from Beam-N. α_R is of more concern, since it moves forward to the high-density side toward the center of an ICF target, causing direct preheating. As can be seen from Fig. 6(a), α_R is smaller than α_L, but it is still of concern, since α_L ∼ 2% is much larger than the ∼0.1% hot-electron preheating that is tolerable for direct-drive schemes.4

Increasing the intensity of either Beam-N or Beam-L is found to increase the hot electron fluxes in the N-L system, as demonstrated by Fig. 6(b). The intensity of Beam-N increases from I_N = 3 × 10¹⁴ W/cm² in case I to I_N = 5 × 10¹⁴ W/cm² in case II, causing α_R to increase from 2% to 6% in the nonlinear stages after t = 5 ps. Increasing the intensity of Beam-L from I_L = 5 × 10¹³ W/cm² in case I to I_L = 1 × 10¹⁴ W/cm² in case V causes α_R to rise from 2% to about 5%, while lowering I_L down to I_L = 3 × 10¹³ W/cm² in case IV reduces α_R to 0.2%. These trends are evidence that strengthening either “amplification” or “seed” in the N-L system will increase hot-electron generation, while weakening either of them will decrease it. It is also shown in Fig. 6(b) that the absence of Beam-L in case III (I_L = 0) completely suppresses TPD and produces no hot electrons, even if I_N is larger than in case I, which shows that the absolute growth driven by Beam-L is an essential requirement for hot-electron generation in this regime. Lowering I_L is proposed as a good strategy for mitigating the hot-electron fluxes in the N-L system, since it does not require very much alteration of the total input laser energy.

In addition to the energy fluxes, the temperature T_hot, which determines the penetration depth, is another key feature of the hot electrons from the target-preheating point of view. In the simulations, T_hot is obtained by Maxwellian fitting of the high-energy tail in the electron distribution function.35T_hot is found to reach a quasi-steady value in the highly nonlinear stage of each case, as can be seen from Fig. 6(c), which also shows that T_hot depends principally on I_N and is not that sensitive to I_L. In the nonlinear stage, T_hot reaches about 40 keV in case I. Increasing I_N to 5 × 10¹⁴ W/cm² (case II) causes T_hot to rise to 50 keV. However, T_hot is found to be very similar (∼40 keV) in cases I and V, in which I_N is kept the same but I_L is quite different. Staged acceleration of hot electrons in case I is demonstrated in Fig. 6(d), which shows that the maximum longitudinal momentum p_x of the hot electrons increases from smaller x to larger x (as indicated by the blue arrow). This staged acceleration process is similar to that in the single-beam normal-incidence regime reported in Ref. 12: the EPWs in the low-density region have low phase velocities that allow effective first-stage acceleration of thermal electrons. The electrons are then stage-accelerated by the EPWs with higher phase velocities in the higher-density region.

The scaling laws obtained through numerical fitting of comprehensive PIC simulations36 have been reported to give good predictions of T_hot for cases with η ≳ 1 in the single-beam normal-incidence regime. Although the two-beam N-L regime with η < 1 is certainly beyond the original scope of Eq. (2) of Ref. 36, this formula is still suitable for predicting T_hot in an N-L regime, since Beam-L can be considered as a small energy perturbation added to the single-beam normal-incidence system including Beam-N only. Equation (2) of Ref. 36 predicts T_hot = 64 keV for case I, which is larger than the simulation value T_hot = 40 keV. Better agreement is obtained for larger η. Lower T_hot poses a lower preheating risk, but the unexpected growth of TPD in the N-L regime when η is well below unity remains an uncertain factor for direct-drive ICF.

TPD driven by two laser beams in an N-L system has been investigated through PIC simulations. It has been shown that in the presence of a low-intensity Beam-L that exceeds its large-angle-incidence TPD threshold (ξ > 1), significant TPD growth can be driven in a regime where η is well below unity. The large angles are particularly important, because the TPD threshold has been found to be acceptably low only within a narrow range of the incident angle θ near 60°.25 TPD collectively driven by Beam-L and Beam-N is expected to occur if θ of Beam-L varies in this narrow range, although at higher thresholds than θ = 60°, whereas TPD is not expected to occur if θ of Beam-L is not large enough (i.e., <50°) given that I_L is very weak. Both beams contribute to the growth of TPD via the “seed-amplification” process in which the absolute instability driven by Beam-L provides the seeds that are convectively amplified by Beam-N. The levels of the seeds are mostly determined by Beam-L, while the amplification factors are determined by Beam-N. The two beams contribute to TPD growth in different ways, both making TPD energetically important and causing significant pump depletion as well as hot-electron generation in the N-L system. The hot-electron fluxes are found to depend on the intensities of both beams, while the hot-electron temperature is found to depend mostly on the intensity of Beam-N.

The low threshold of TPD in this regime can be harmful to direct-drive ICF and imposes extra restrictions on target design, since TPD is able to grow significantly in previously unexpected regions. Under experimental conditions, Beam-L comes from a side portion of the laser beam that grazes the edge of the corona plasma. Therefore, to lower the risk of TPD, it would be a good strategy to shrink the width of each laser spot, which is already well known to benefit implosion performance by mitigating CBET.37–39 Reducing the spot size of the laser beams can effectively lower I_L in the boxed zone illustrated in Fig. 1(a), thereby mitigating two-beam TPD and representing another advantage of direct-drive ICF.

The finite-spot-size effect in the N-L system, especially if Beam-L has a narrow spot size (e.g., a speckle) deviating from the plane-wave approximation used in the present simulations, is an important topic to be studied in the future. It has been shown that a single large-angle-incidence laser with a small spot size can still stimulate TPD at a very low laser intensity, but the TPD growth and hot-electron generation are weaker than in the case of a plane wave.25 On the basis of the single-beam results, it can be speculated that the low TPD threshold of the N-L system still exists with finite-spot-size beams, although the pump depletion and hot-electron generation can be quantitatively reduced. Laser speckle in LPI is another important effect that has attracted intensive research interest in both simulations36,40–42 and experiments.43 Recent experimental work has revealed an enhancement of TPD activity as well as a lower TPD threshold with speckled beams, owing to the above-average intensities in the speckles.43 This suggests that a speckled geometry may even lower the TPD threshold in the N-L system.

Acknowledgment. We thank the UCLA-IST OSIRIS Consortium for the use of OSIRIS. This research was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 12375243 and 12388101, by the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant Nos. XDA25050400, XDA25010200, and XDA25010100, by the Science Challenge Project, and by the Fundamental Research Funds for the Central Universities. The numerical calculations in this paper were performed on the supercomputing system in the Supercomputing Center of the University of Science and Technology of China.