The nonlinear physics of crossbeam energy transfer (CBET) for multispeckled laser beams is examined using largescale particleincell simulations for a range of laser and plasma conditions relevant to indirectdrive inertial confinement fusion (ICF) experiments. The timedependent growth and saturation of CBET involve complex, nonlinear ion and electron dynamics, including ion trappinginduced enhancement and detuning, ion acoustic wave (IAW) nonlinearity, oblique forward stimulated Raman scattering (FSRS), and backward stimulated Brillouin scattering (BSBS) in a CBETamplified seed beam. Iontrappinginduced detuning of CBET is captured in the kinetic linear response by a new δfGaussianmixture algorithm, enabling an accurate characterization of trappinginduced nonMaxwellian distributions. Ion trapping induces nonlinear processes, such as changes to the IAW dispersion and nonlinearities (e.g., bowing and selffocusing), which, together with pump depletion, FSRS, and BSBS, determine the timedependent nature and level of CBET gain as the system approaches a steady state. Using VPIC simulations at intensities at and above the onset threshold for ion trapping and the insight from the timedependent saturation analyses, we construct a nonlinear CBET model from local laser and plasma conditions that predicts the CBET gain and the energy deposition into the plasma. This model is intended to provide a more accurate, physicsbased description of CBET saturation over a wide range of conditions encountered in ICF hohlraums compared with linear CBET gain models with ad hoc saturation clamps often used in laser raybased methods in multiphysics codes.
I. INTRODUCTION
Laserplasma instabilities (LPI) reduce lasertarget coupling and are fundamental limiters of fusion performance for laserdriven inertial confinement fusion (ICF) approaches.^{1,2} In order to build on the success of the recent demonstration of ICF ignition,^{3} we need to understand the nonlinear dynamics and saturation of LPI and develop an improved capability for modeling LPI effects on ICF experiments. Major LPI concerns include stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS), where the laser light scatters off an electronplasma (Langmuir) wave and an ionacoustic wave (IAW), respectively. Crossbeam energy transfer (CBET) is another LPI process by which energy from overlapping laser beams is transferred from the higher to the lower frequency beam through the excitation of resonant IAW.^{4}
Significant efforts have been made to include LPI effects in radiationhydrodynamics (radhydro) codes using linear models^{5–7} with demonstrated success in improving agreement between modeling and experiments on the NIF.^{8} These models are based on wellestablished linear theory. However, linear physics alone may not adequately describe the complex, nonlinear behavior of LPI at length and time scales relevant to the experiments.^{9–12} In particular, in order to match experimental observables, a common practice is to include empirically obtained ad hoc CBET saturation clamps on IAW amplitudes to reduce CBET in linear models in radhydro codes. These clamps use varying levels of electron density fluctuations associated with IAW, ranging from $ \delta n e / n e$ of order $ 10 \u2212 4$ to $ 10 \u2212 2$ depending on experimental platforms.^{8,13,14} Some of the clamp values used are too small to be explained by known CBET saturation mechanisms. This practice speaks directly to the need to incorporate nonlinear physics into the modeling of CBET, a point also raised by, e.g., Turnbull et al.^{15}
Moreover, approximations are often used to make the linear theory tractable, such as treating light as plane waves (speckle effects neglected), using steadystate coupledwave equations in the strong damping limit (advection of plasma waves neglected and unable to treat plasma responses in weakly damped regimes), or using a paraxial approximation (limited light propagation angles) or a fluid treatment (kinetic effects neglected). Improvements have been made recently. For example, kinetic descriptions have been included in the linear theory.^{16} Also, raytracing using complex geometrical optics techniques have been employed that allow for the modeling the effects of speckles^{6} (though computationally more expensive). Methods have been developed that do not limit light propagation angles.^{17,18} However, these improvements fall short of providing a satisfactory treatment of the nonlinear LPI associated with CBET saturation.
To retain fully kinetic electron and ion physics of LPI, the particleincell (PIC) plasma simulation method, together with highperformance computing, is a powerful tool for understanding the nonlinear saturation of LPI and for developing physicsbased saturation models to eliminate the need for ad hoc saturation clamps. This approach includes weakly damped regimes excluded in current linear LPI models. Improving the modeling of CBET in multiphysics codes for modeling ICF experiments has been identified by the ICF community as one of the key outstanding questions in LPI research.^{19} Recent capsule implosions on the NIF exceeded Lawson's criterion for ignition^{3} and have achieved net fusion energy production.^{20} Future experiments on the NIF will seek to increase fusion yield to 10 MJ, which will require possibly higher laser energy and intensity and an improved predictive nonlinear LPI modeling capability. In this paper, we explore how PIC simulations performed in relevant parameter spaces can be used to develop a prototype nonlinear LPI model for ICF design codes.
In this work, the nonlinear dynamics and saturation of CBET are examined using largescale VPIC^{21–24} simulations in twospatial dimensions (2D) for a range of plasma and laser conditions. These include various plasma densities, temperatures, laser beam intensities, crossing angles, and beam diameters. In the simulations, nonlinear effects resulting from particle trapping for both ions and electrons, and secondary instabilities such as oblique forward stimulated Raman scattering (FSRS) and backward stimulated Brillouin scattering (BSBS) are captured, as we discuss below. Using these simulations, our research goal is to develop a physicsbased nonlinear CBET model that includes saturation from these nonlinear physics processes. The model is to be implemented as a subgrid model in multiphysics codes to couple CBET effects to improve the modeling of ICF experiments.
In the remainder of the paper, following a description of the simulation setting (Sec. II), we discuss the underlying processes contributing to the timedependent nonlinear saturation of CBET (Sec. III), including effects of ion trapping and secondary instabilities. In particular, we developed a new, automated, δfGaussianmixture algorithm to represent nonMaxwellian distribution functions from ion trapping and employed the method to evaluate the timedependent plasma response resulting from ion trapping. We explain the effects of these nonlinear processes on CBET in the context of prior studies and how they are diagnosed in VPIC simulations in the parameter space we modeled in this work. In the presence of these nonlinear effects, CBET saturation is a complex problem that depends on details of the laser and plasma parameters. Our goal here is not to provide detailed analyses of specific special cases where one of these saturation mechanisms dominates over the others or quantitative analyses of the degrees to which each of these nonlinear effects contributes to the CBET saturation process. Such kinetic details, though important and worthy of future examination, are beyond the scope of this work. Rather, we present the nonlinear scaling of systemintegrated CBET gain vs key laser and plasma parameters (Sec. IV) intended for our nonlinear CBET models and discuss how these nonlinear behaviors may be different from linear theory expectations based on the linear plasma response function. Using these simulation results at intensities at and above the onset threshold for ion trapping and the insight from the timedependent saturation analyses, a nonlinear CBET model is constructed (Sec. V) to account for the gain and the energy deposition into the plasma for feeding back to hydrodynamics in design codes. Finally, we provide discussions on other effects not included in the CBET model (Sec. VI) and a summary of our main findings (Sec. VII).
We note that to the best of our knowledge, this work represents the first attempt to employ results from large suite of largescale PIC simulations to construct a nonlinear CBET model for the modeling of ICF experiments in multiphysics codes. The simulations presented here are at intensities near or above the onset threshold intensity for ion trapping. We show that nonlinear effects are, in general, small near the onset threshold intensity but increase with intensity. If our nonlinear CBET gain results are to be compared with those from the linear CBET model, one would expect the disagreement to be small near the onset threshold intensity but to increase with intensity. A thorough quantitative comparison of linear CBET gains with VPIC simulations is outside of scope since there exist many forms of linear CBET theory. We point out, however, one particular form of linear theory CBET gain was compared with VPIC simulations (including iontrappinginduced nonlinear effects), and the results showed that linear gain exponent is up to 2× higher than the nonlinear saturated gain.^{25} To put the results of our work in the context of the CBET linear theory frameworks employed in raybased models, in Sec. V, a subset of the VPIC simulations and the nonlinear CBET model are compared with linear predictions using the same form of linear CBET theory as in Ref. 25. The discrepancies underscore the need to improve linear CBET models when nonlinear effects are important. Interested readers will be able to use our results in the form of either the CBET gain shown in the figures or gain expressed in closed forms as a function of laser and plasma conditions to compare with their choice of linear CBET model. One can also use the nonlinear CBET model expressions to estimate energy transfer in designing focused CBET experiments.
II. SETUP OF SIMULATIONS
The simulations are performed in two spatial dimensions (2D) in the (x, z) plane with cell size set to be close to the Debye length λ_{D} in the x and z directions. We use 512 particles per cell per species, generally. The multispeckled laser beams have wavelength $ \lambda 0 = 351$ nm, an average intensity $ I ave$, and a flattop temporal pulse shape with rise time of 0.25 ps to $ I ave$ (this rise time gives the same instability growth as that from the use of a longer rise time). The laser polarization is in the y direction and its field E_{y} is specified at the launch boundary in a manner that approximates a Gaussian random field,^{26–28} which in vacuum creates a random distribution of F/8 speckles with characteristic width $ 1.2 F \lambda 0 = 3.4$ μm and length $ 2 \pi F 2 \lambda 0 = 141$ μm. Absorbing boundary conditions are used for the fields.
The plasma has electrons and He^{2+} ions with a uniform density and initial Maxwellian distributions, generally. (NonMaxwellian initial distributions are considered in Sec. VI D.) The CBET behavior is explored in simulations performed at a range of plasma densities from $ n e / n cr = 0.01$ to 0.06 where $ n cr$ is the critical density, with an electrontoion temperature ratio $ T e / T i = 2$ and 4 (T_{e} = 3 keV). We use a range of laser beam average intensities, from $ I ave = 7.5 \xd7 10 13$ to $ 1.2 \xd7 10 15$ W/cm^{2}, with beam diameters 140, 280, and 420 μm, and beam crossing angles $ \theta = 15.3 \xb0 , \u2009 31.0 \xb0 , \u2009 47.0 \xb0 , \u2009 and \u2009 99.0 \xb0$.
More than 60 largescale VPIC simulations have been used to produce the results we discuss in this paper. The majority of the simulations have been performed at intensity $ I ave = 3.0 \xd7 10 14$ W/cm^{2} at various laser and plasma conditions. A collection of simulations at $ I ave < 3.0 \xd7 10 14$ W/cm^{2} have been performed to determine the onset threshold for ion trapping for various plasma conditions, above which the nonlinear model applies. Simulations have also been performed at higher intensity $ I ave = 4.5$ and $ 6.0 \xd7 10 14$ W/cm^{2} to explore the model dependence on intensity. Additionally, one simulation result for $ I ave = 12 \xd7 10 14$ W/cm^{2} is shown in Fig. 5(b) at low plasma density where FSRS and BSBS are absent to examine how ion trapping and its detuning effects increase with intensity. The values of $ I ave$ we examined allow for the application of our model to existing ICF facilities as well as for evaluating CBET risk on future facility upgrades. The results of systemintegrated CBET gain vs time shown in the figures are sampled from a diverse range of laser and plasma conditions to illustrate the complexity of the nonlinear CBET behavior. The resulting nonlinear CBET model is expressed as a function of laser intensity.
The majority of the simulations in this work are collisionless and in the absence of plasma flow with frequency chosen for matching conditions as discussed below. In simulations, “Maxwellian refluxing” boundary conditions are used for the particles: when a particle encounters a boundary, it is destroyed and a new particle is injected with velocity sampled randomly from a Maxwellian distribution at fixed temperature equal to the initial plasma temperature.
The frequency and wavenumber matching conditions satisfy $ \omega = \omega p \u2212 \omega s$ and $ k = k p \u2212 k s$ where subscripts “p” and “s” stand for pump and seed, respectively, and ω and $k$ are for the IAW. In the simulations, the difference of the laser pump and seedbeam frequencies is set to that of the IAW which is obtained from the kinetic linear theory at wavenumber $ k \lambda D = 2 ( k 0 \lambda D ) \u2009 sin \u2009 ( \theta / 2 )$, where $ \lambda D$ is the Debye length, k_{0} is the laser wavenumber, and θ is the beam crossing angle as indicated in Fig. 1. The light wave $ k 0 \lambda D$ is determined from the dispersion relation $ k 0 \lambda D = ( v th , e / c ) ( \omega 0 / \omega pe ) 2 \u2212 1$, where $ v th , e = T e / m e$ is the electron thermal speed, $ \omega pe = ( 4 \pi n e e 2 ) / m e$ is the electron plasma frequency, and T_{e}, n_{e}, e, and m_{e} are the electron temperature, density, charge, and mass, respectively. Thus, the IAW $ k \lambda D$ is determined from the beam crossing angle and plasma conditions.
A representative simulation geometry is shown in Fig. 1 where the pump and seed beams cross at angle θ and propagate at $ \theta / 2$ with respect to $ \xb1 x \u0302$ direction, launched from the left boundary; the IAW propagates in the downward $ \u2212 z \u0302$ direction. For simulations at larger θ (e.g., 90°), the seed beam is launched from the right boundary and the pump from the top boundary. We define the CBET gain percentage in the seed beam as $ gain \u2009 ( t ) = 100 % \xd7 [ P s \u200a out ( t ) \u2212 P s \u200a in ( t ) ] / P s \u200a in ( t )$, where $ P s \u200a in ( t )$ and $ P s \u200a out ( t )$ are the seed beam input and output power, respectively. The timeaveraged gain is defined as $ \u222b 0 T gain ( t ) d t / time ( t )$, where $ time ( t )$ is the instantaneous time and T is the total interaction time. In the simulations, CBET gain can be affected by secondary instabilities, such as FSRS and BSBS, that become unstable in the amplified seed beam,^{29,30} as illustrated in Fig. 1.
SRS and SBS grow from thermal noise inherent in the PIC method. For 512 particles per cell per species we used, both Langmuir wave thermal fluctuation levels and Thomson scatter in simulations were examined and shown to be close to the values estimated in ICF hohlraum plasmas.^{27} Moreover, noise sensitivity for FSRS was examined in our prior CBET simulations^{29} where it is found that FSRS level is comparable for 512 and 4000 particles per cell per species. In modeling the recent CBET experiments,^{31,32} we simulated pump beam stability for FSRS using VPIC simulations. At the low plasma density for those experiments, the pump laser beam is stable to FSRS in our simulations, in agreement with experimental observations.^{33} So it is unlikely that our results presented in this work contain numerical artifacts of concern.
Although the majority of the simulations are collisionless and without plasma flow and are initiated with Maxwellian distributions of electrons and ions, select simulations have been performed using a binary collision model^{34} or with the presence of flow to examine their effects on CBET processes. These are discussed in Secs. VI A and VI B. The effects of initial superGaussian electron distribution on CBET dynamics as inferred from experiments^{31,32} are also examined in Sec. VI D.
III. TIMEDEPENDENT NONLINEAR SATURATION OF CBET
In this section, we discuss the spatial and timedependent nature of CBET gain influenced by iontrappinginduced nonlinear effects and secondary instability FSRS and BSBS. Together with pump depletion, they determine the saturation level of CBET. Each of these saturation mechanisms has been examined in prior studies and we indicate them below in relevant contexts. Our focus here is to discuss their effects on the CBET saturation in the parameter space we modeled and explain how these processes are diagnosed in VPIC simulations.
A. Iontrappinginduced CBET detuning: Changes in the linear kinetic plasma response function from trappingmodified nonMaxwellian ion distribution
One of the key nonlinear processes determining the overall CBET gain is the spatial and timedependent ion trapping in the beam overlap region. From the linear theory, CBET is sensitive to the IAW damping rate, and the energy transfer is proportional to the imaginary part of the plasma response function $ \Gamma ( \omega , k ) = \chi e ( \omega , k ) [ 1 + \chi i ( \omega , k ) ] / [ 1 + \chi e ( \omega , k ) + \chi i ( \omega , k ) ]$,^{16,35} where $ \chi j ( \omega , k ) = \u2212 \zeta \u2032 / ( 2 k 2 \lambda D j 2 )$ are the Vlasov susceptibilities from plasma species j, $ \zeta j \u2032 ( \omega , k ) \u2261 Z \u2032 [ \omega / 2 k v t h , j ]$, and $ Z \u2032$ denotes the derivative of the plasma dispersion function^{36} with respect to its argument.
We examine the changes of $ Im \u2009 \Gamma ( \omega , k IAW )$ evaluated at the resonant IAW wavenumber using trappingmodified nonMaxwellian ion distribution from the Gaussianmixture method, contrasting with that from the use of an initial singleMaxwellian distribution. The reduction in the value of $ Im \u2009 \Gamma ( \omega , k IAW )$ at the initial matching frequency allows one to infer the detuning of CBET resulting from ion trapping.^{25}
Figures 2(a) and 2(b) show strong and weak trapping cases, respectively, at $ v ph / v th , i = 3.38$ and IAW $ k \lambda D = 0.3$ (for $ \theta = 47 \xb0$, $ n e / n cr = 0.04$, and $ T e / T i = 4$). They are sampled from different spatial locations in the speckled laser beams for simulations at different intensity. In Fig. 2(a), the 2D distribution function is from a simulation with gain indicated by the dashed curve in Fig. 5(a1) (to be discussed in detail in Sec. IV C) at saturation during the steadystate phase of gain (the time is indicated by the solid circle on the dashed curve). The trappingmodified 1D reduced distribution is represented accurately by a summation of Gaussians (a mixture of 27 Gaussians is shown here). Modification of the ion distribution due to trapping leads to a reduction of IAW damping, as evidenced by the narrow width and enhanced peak of the solid curve of $ Im \u2009 \Gamma ( \omega , k IAW )$. It also results in a shift of the peak of the function toward lower frequency and a reduction of $ Im \u2009 \Gamma ( \omega , k IAW )$ at the resonant IAW ω, i.e., the frequency corresponds to the peak of the dashed curve. In this example, the value of $ Im \u2009 \Gamma ( \omega , k IAW )$ for the solid curve at the driving frequency (which is the frequency corresponding to the peak of the dashed curve) is reduced by 6× compared with that of the dashed curve based on the initial singleMaxwellian distribution. This detunes the CBET coupling at the driving frequency and wavenumber, which are fixed in time. However, we note that the energy transfer from the pump to the seed beam may still be possible (though maybe less efficient) provided the plasma response function at the matching frequency is nonzero. One can thus view the iontrappinginduced detuning process as a partial disruption of the resonant coupling at the driving frequency and wavenumber. Note that ion trapping is space and timedependent, even when the systemintegrated CBET gain appears to reach a steadystate. The results in Fig. 2 have been obtained in a small volume of the large beam crossing region at a fixed time. Even if the coupling function vanishes at one point in space and time in the crossing region, provided it is nonzero elsewhere in the region or at other times, there may still be CBET.
Frames in Fig. 2(b) are for a weaker iontrapping case (obtained at a spatial location where the trapping modification of the ion distribution is weaker) with gain indicated by the solid curve in Fig. 5(a1) at saturation during the steadystate of gain (the time is indicated by the solid circle). The trappingmodified 1D distribution function is described by 8 Gaussians. The value of $ Im \u2009 \Gamma ( \omega , k IAW )$ at the driving frequency for the solid curve is reduced by 3× compared with that for the dashed curve based on the initial singleMaxwellian distribution. This case shows that even a small trapping tail is effective in detuning CBET coupling at the driving frequency and wavenumber.
Another strong trapping case is presented in the frames of Fig. 2(c) at $ v ph / v th , i = 3.15$ and IAW $ k \lambda D = 0.61$ (for $ \theta = 47 \xb0$, $ n e / n cr = 0.01$, and $ T e / T i = 4$). The 2D distribution function is from a simulation with gain indicated by the dashed curve in Fig. 5(b) at a time (indicated by the solid circle on the dashed curve) when the gain saturates and drops to a steadystate value. Its 1D reduced form is represented by 10 Gaussians. The value of $ Im \u2009 \Gamma ( \omega , k IAW )$ at the driving frequency is significantly reduced by over 460× from the peak of the dashed curve based on the initial singleMaxwellian distribution. In this case, the trappinginduced frequency downshift is large. The CBET gain indicated by the dashed curve in Fig. 5(b) rises to its peak value (near 30%) and then drops to below 20%, approaching a steadystate value. We note that in all of our simulations, provided that they are run for a long enough time, the systemintegrated CBET gain is found to eventually reach a steady state value. This is because in the simulation system, energy from the laser beams enters and exits the boundaries and particle and wave energy from the LPI processes also exit the boundaries, which act as an energy sink. Trapped particles in a spatially localized region propagate to nearby regions. Trapped particles also propagate out of the overlap region and are replenished by particles flowing into the overlap region from the boundaries. In addition, other nonlinear effects from IAW nonlinearities and secondary instabilities in local regions may fluctuate and evolve with time. At a longer timescale (typically after a few tens of ps in our simulations), the systemintegrated gain settles to a steadystate value (as well as the power into ions shown in Fig. 10), however, “quasisteadystate” may be a more appropriate description for the overall system given the local fluctuations of nonlinear processes. (Note, given the finite computing resources available to us, while we were able to run all our VPIC simulations sufficiently long in time to capture the main nonlinear effects on CBET gain, in a subset of our simulations, we were not able to run until a true steadystate was reached in the gain percentage. In all cases, we used the timeaveraged gain in constructing our subgrid nonlinear CBET models for radhydro codes.)
Finally, we discuss a case in Fig. 2(d) where $ v ph / v th , i = 2.815$ and IAW $ k \lambda D = 0.018$ (for $ \theta = 31 \xb0$, $ n e / n cr = 0.05$, and $ T e / T i = 2$) with the CBET gain shown by the green curve in Fig. 6(b2) (Sec. IV D). Here, the IAW phase velocity is closer to the bulk of the distribution with small but sharp trapping modification sampled at a time when the gain rises to a steadystate value (indicated by the solid circle on the green curve). Its 1D reduced form is described with 11 Gaussian functions. The value of $ Im \u2009 \Gamma ( \omega , k IAW )$ at the driving frequency is reduced by over 5× from the peak of the dashed curve based on the initial singleMaxwellian distribution. In this case, and similarly for other cases (not shown here) in which the IAW phase velocity is close to the bulk of the distribution, we found that the trappinginduced frequency downshift is small, but the detuning at the driving frequency and wavenumber can be effective.
We emphasize the spatial and timedependent nature of ion trapping effects on CBET gain. In situ iontrapping modification of the distribution varies with local speckle intensities of the crossing beams and the timedependent average intensity of each beam resulting from both pump beam depletion and seed beam amplification. [For example, trapping in Fig. 2(a) from a simulation at lower intensity can be stronger than trapping in Fig. 2(b) at a higher intensity since they are sampled at different local regions and times.] The trapping modification of the distribution and CBET gain in local regions evolve in time even after the integrated gain of the system reaches a steadystate value. Since the pump laser beams consist of collections of speckles, i.e., spatially localized intensity structures, pump depletion can be space and timedependent as well.
It is also important to note that in all the above cases, in the process of the peak of $ Im \u2009 \Gamma ( \omega , k IAW )$ shifting to a lower frequency, its value also increases sharply as a result of damping reduction. This can lead first to an enhancement of the gain and then to a reduction of the gain. Together with other nonlinear processes discussed below in Secs. III B and III C, trappinginduced detuning affects the overall steadystate gain level, independent of how the gain curve approaches steadystate, whether rising to a peak steadystate value or increasing to a peak then dropping to a steadystate value as shown in Fig. 5 (Sec. IV C).
B. Other iontrappinginduced nonlinear effects on CBET saturation: IAW wavefront bowing, breakup and selffocusing, and speckle interaction
In addition to CBET detuning from trappingmodified distribution and plasma response functions, ion trapping is known to contribute to SBS saturation from IAW wavefront bowing and subsequent wavefront breakup and selffocusing, as observed in 2D and 3D simulations in speckle geometry (for $ n e / n cr = 0.036$ with T_{e} = 700 eV and T_{i} = 140 eV in He^{2+} plasma).^{40,41} The IAW wavefront bowing is due to a trapped particle nonlinear frequency shift,^{42–44} which grows with wave amplitude. It has been speculated (though not proven)^{40} that IAW wavefront breakup and selffocusing may be caused by trapped particle modulational instability (TPMI) and proceeds in a manner similar to electron plasma wave (EPW) wave selffocusing for SRS.^{40,41,45,46}
In these prior studies, EPW selffocusing/filamentation was found to lead to transverse EPW phase variation, increased sideloss of trapped electrons, and a subsequent decrease in wave energy, and termination of the SRS pulse. The growth of transverse modes from TPMI and the subsequent nonlinear evolution that leads to EPW selffocusing were examined using a particular family of Bernstein–Greene–Kruskal (BGK) equilibria^{47} treating largeamplitude waves in theory^{45,46} and in 2D PIC simulations in a periodic, onewavelength long system in the absence of wavefront bowing.^{40,41} As the wave amplitude exceeds the TPMI threshold, the wave selflocalizes transversely, leading to velocity diffusion by transverse modes. Loss of trapped particles through the sides of the narrow, elongated speckle structures (sideloss) increases and the wave energy decreases as the particle kinetic energy increases (see Fig. 9 in Ref. 41). Thus, in this picture, the wave damping is amplitude dependent and bowing may not be required for EPW selffocusing. Selffocusing of EPW in the absence of bowing was examined in a number of other studies^{48–51} using theory, Vlasov and PIC simulations where the results shown are in general agreement with Refs. 45 and 46. The decay of the wave energy associated with the sideloss of particles was also examined in kinetic simulations.^{51} The interplay between bowing and EPW selffocusing/filamentation was examined in 2D and 3D kinetic simulations^{52} of both f/4 and f/8 speckles. This study showed that in f/4, with smaller plasma volumes and larger transverse gradients, bowing dominated; in f/8, EPW selffocusing dominated. A recent study using the rayincell method also showed transverse modes dynamics resulting from bowing and its effects on SRS.^{53}
Similarly for IAW, kinetic simulations show that wavefront bowing and transverse modes associated with breakup can effectively dissipate wave energy and increases wave damping, contributing to SBS and CBET saturation. IAW bowing and breakup are found to contribute to CBET saturation in VPIC simulations with multispeckled laser beams (for $ n e / n cr = 0.04$ with T_{e} = 3 keV and $ T i = T e / 4$ in He^{2+} plasma)^{29} and in collisional VPIC simulations of CBET experimental setting^{31,32} at low plasma density (in fully ionized hydrogen and nitrogen plasma with $ n e / n cr = 0.006$, T_{e} = 600 eV, T_{i} = 150 eV, and $ n e / n cr = 0.012$, T_{e} = 840 eV, and T_{i} = 130 eV).^{25} Indeed, in Ref. 25, decrease in wave energy associated with CBET IAW breakup was shown from VPIC simulations. With respect to the origin of the IAW breakup, extending the approach for EPW selffocusing but with a source of IAW instead of EPW would be an interesting followon study. Though relevant to CBET saturation, this is a separate topic that is outside the scope of our paper. However, it is important to note that the nonlinear CBET model we constructed from the VPIC simulations does include contributions from IAW bowing and wavefront breakup regardless of the exact origin of IAW selffocusing.
The IAW wavefront bowing and breakup can be diagnosed by examining the structures in the electrostatic field, as in Refs. 25, 40, and 41, or the structures in the ion velocity (first moment of the ion distribution function on the grids), as in Ref. 29. The IAW amplitudes are larger in regions of the highintensity speckles in the beam overlapping region, with their structures evolving from being coherent initially to breaking up to smaller scale structures in the direction transverse to the IAW propagation. Resonant ions can be tracked with tracer particles, and their orbits can be followed in space and time, exhibiting trapping bounce motions in largeamplitude IAW and detrapping as trapped ions pass through the regions where the IAW breakup occurs (see Fig. 2 in Ref. 29).
The crossing regions of the speckled laser beams in this work comprise a large number of overlapping speckles and speckle interaction^{27,54–57} has important effects on CBET. Transport of ions that resonantly interact with IAW in strong speckles can seed and enhance the growth of CBET in neighboring speckles by reducing damping. This speckle interaction can lead to nonlocal coupling and an increase in the wave coherence length to a scale length much larger than the speckle widths, enhancing energy transfer.
C. Effects of secondary instabilities FSRS and SBS on CBET saturation
Since nonlinear CBET gain depends on the in situ intensity of both the pump beam (which may be affected by pump depletion) and the seed beam (which may be amplified), FSRS and BSBS in the seed beam during energy transfer are important to the CBET dynamics. In our prior work,^{29} it was found that CBET saturation can involve excitation of oblique FSRS in the amplified seed beam, which leads to beam deflection at a large angle from the initial beam propagation direction and a frequency downshift (by the plasma frequency). FSRS can lead to hot electrons with energy exceeding 300 keV, and its saturation can occur on fast ∼ps timescales by EPW selffocusing and can lead to enhanced sideloss of hot electrons. The presence of a density gradient reduces the growth of FSRS. FSRS may deflect more than 40% of the incident seed beam energy and may convert a few percent of incident beam energy into hot electrons. Also in our prior work,^{30} it was found that at an increased beam path length, BSBS may become unstable in the CBET amplified seed beam and contribute to the saturation of CBET. Its effects reduce with temperature ratio $ T e / T i$.
FSRS and BSBS play similar roles in CBET dynamics in the present work and are two additional causes of the timedependent nature of CBET gain. We will discuss the effects of FSRS and BSBS on the CBET gain in simulations presented in later sections. FSRS can be diagnosed from (1) its modification to the electron distribution, (2) the daughter EPW structure in electrostatic field, and (3) the hot electron spectrum that tallies the flux of particles leaving the system.^{27} The presence of BSBS in the simulations can be detected by three methods: (1) the seed beam flux is measured on the injection boundary, and reduction of this flux over time may indicate reflectivity due to BSBS. (2) The FFT power spectrum of laser field E_{y} along the line of the beam injection boundary may show a downshifted frequency component by an amount consistent with that of the BSBS daughter IAW. (3) Modification in the ion distribution function consistent with characteristics of BSBS daughter IAW. Examples of these diagnostics can be found in Fig. 13 in Ref. 30.
It is found that the presence or absence of FSRS in the CBET amplified beam is a strong function of density, in addition to laser intensity, and that FSRS is negligible for low density plasma (e.g., $ n e / n cr \u2009 0.01$). The amount of BSBS in the amplified seed beam increases with density, temperature ratio $ T e / T i$, and laser intensity.
IV. NONLINEAR SCALING OF CBET GAIN VS PLASMA AND LASER CONDITIONS
In this section, we explore the nonlinear behavior of CBET in the presence of ion trapping and secondary instabilities. In particular, we examine the scaling of the gain with various plasma and laser conditions, namely, IAW $ k \lambda D$, temperature ratio $ T e / T i$, intensity $ I ave$, density $ n e / n cr$, beam crossing angle θ, and beam diameter d. Note that both d and θ determine the interaction length $ L = d / \u2009 sin \u2009 \theta $. As we have discussed, CBET linear gain is proportional to the plasma response function, which reflects characteristics of the IAW dispersion. To put simulation results in the context of linear theory expectations, the CBET gain from the simulations is presented together with the plasma response function as a reference to the linear behavior. These scalings are used to construct a nonlinear CBET model in Sec. V.
In this section, for each simulation presented, we identify one or more of these nonlinear effects contributing to the saturation of the systemintegrated CBET gain as a function of key parameters intended for building our nonlinear CBET models (Sec. V). In the presence of these nonlinear effects, the physics of CBET saturation is complex, with details that depend on plasma and laser conditions. We do not attempt to differentiate here the relative contribution of each nonlinear effect to the CBET saturation.
A. CBET gain depends sensitively on IAW $ k \lambda D$
IAW $ k \lambda D$ is one of the key parameters determining the CBET dynamics, and we use the case at $ n e / n cr = 0.04$ and $ T e / T i = 4$ to show how CBET gain can decrease significantly from increasing $ k \lambda D$. With $ \theta = 47 \xb0$, the resonant IAW $ k \lambda D = 0.30 , \u2009 v ph / v th , i = 3.38$, and the IAW frequency and the damping rate is $ \omega r / \omega pe = 0.005 \u2009 91$ and $ \omega i / \omega pe = \u2212 0.000 \u2009 26$, respectively. With $ \theta = 99 \xb0 , \u2009 k \lambda D = 0.57 , \u2009 v ph / v th , i = 3.18 , \u2009 \omega r / \omega pe = 0.010 \u2009 60$, and $ \omega i / \omega pe = \u2212 0.000 \u2009 64$. Figure 3 shows the imaginary part of the plasma response function $ Im \u2009 \Gamma ( \omega , k IAW )$ at resonant wavenumber [frame (a)] and the CBET gain at intensity $ I ave = 6.0 \xd7 10 14$ W/cm^{2} for $ k \lambda D = 0.30$ (solid) and $ k \lambda D = 0.57$ (dashed) [frame (b)]. The IAW damping rate for $ k \lambda D = 0.57$ is higher (by 2.5×), and the function $ Im \u2009 \Gamma ( \omega , k IAW )$ is broader than that for $ k \lambda D = 0.30$. The peak value of $ Im \u2009 \Gamma ( \omega , k IAW )$ and timeaveraged CBET gain for $ k \lambda D = 0.57$ is 6.4 and 2.7× lower than that for $ k \lambda D = 0.30$, respectively. In both simulations, ion trapping is strong and oblique FSRS becomes unstable in the CBET amplified seed beam. However, they exhibit very different CBET gain profiles as a function of time depending on the IAW damping rate. With smaller IAW damping at $ k \lambda D = 0.30$, as the CBET gain rises in time, FSRS becomes unstable in the seed beam, leading to a gain drop (at around 6 ps). Later in time, CBET continues to grow to a high level and saturates at a steadystate value. With the increased IAW damping at $ k \lambda D = 0.57$, the gain rises to a peak value but starts to drop as FSRS grows in time, approaching a much lower steadystate value. In both cases, iontrapping induced detuning also determines the CBET level.
We note that the two simulations shown here differ in crossing angle and simulation box size. The times required for the beams to cross the simulation domain to overlap and for CBET gain to be diagnosed from the simulation boundaries are different in the two simulations. The relevance of comparing the different IAW damping rates and the plasma response functions in the two simulations is in understanding the systemintegrated CBET gain behavior after the onset of the nonlinear effects rather than the early rise of CBET gain curves. Also, FSRS in the seed beam grows from noise in the simulations and may reach a high level near the seed beam's exiting boundary due to its convective nature. Although FSRS is present in both simulations, the path lengths of the laser beams are different, so the gain lengths for FSRS are different. It is not meaningful to compare the levels of FSRS quantitatively in the two simulations.
B. CBET gain depends sensitively on temperature ratio $ T e / T i$
Another key parameter for CBET is the temperature ratio $ T e / T i$ which relates to electron Landau damping of the IAW, and we use the case for $ n e / n cr = 0.04 , \u2009 \theta = 47 \xb0$, and IAW $ k \lambda D = 0.30$ to illustrate how CBET gain can drop significantly from reducing $ T e / T i$. With $ T e / T i = 4$, the IAW phase speed $ v ph / v th , i = 3.38$, and the IAW resonant frequency and the damping rate are $ \omega r / \omega pe = 0.005 \u2009 91$ and $ \omega i / \omega pe = \u2212 0.000 \u2009 26$, respectively. With $ T e / T i = 2 , \u2009 v ph / v th , i = 2.77 , \omega r / \omega pe = 0.006 \u2009 85$, and $ \omega i / \omega pe = \u2212 0.000 \u2009 89$. The kinetic solutions for $ Im \u2009 \Gamma ( \omega , k IAW )$ are shown in Fig. 4(a), and the CBET gain from simulations at $ I ave = 3.0 \xd7 10 14$ W/cm^{2} are shown in Fig. 4(b). The IAW damping rate for $ T e / T i = 2$ is 3.4 times higher than that for $ T e / T i = 4$, while the peak value of $ Im \u2009 \Gamma ( \omega , k IAW )$ and timeaveraged CBET gain at $ T e / T i = 2$ are 3.4 and 4.8 times lower than that at $ T e / T i = 4$, respectively. In the simulation with $ T e / T i = 4$, CBET grows to a steadystate value with the presence of strong ion trapping and FSRS in the CBET amplified seed beam. With increased IAW damping rate at $ T e / T i = 2$ in the presence of moderate ion trapping and the absence of FSRS, CBET increases to its peak value and then drops to a much lower steadystate value compared with that for $ T e / T i = 4$. The latter case has a smaller $ v ph / v th , i$ value, and the IAW can therefore interact with and Landau damp on more ions whose velocities are closer to those of the bulk of the distribution than those in the former case. These results show again that CBET gain can exhibit very different timedependent profiles depending on the IAW damping rate and the wave phase velocity with respect to the ion thermal velocity. We note that although iontrapping modification of the distribution function may detune the CBET coupling at the driving frequency and wavenumber at one location, the energy transfer from the pump to the seed beam can occur at other locations where the plasma response function is nonzero. The systemintegrated CBET gain is a collective response of the spatial and timedependent local processes. How much the ion trapping affects the systemintegrated CBET gain depends on the IAW damping of the plasma, as shown by the results here.
C. Effects of laser intensity on CBET gain and threshold intensity for nonlinear behavior
Laser beam intensity is another controlling parameter that determines the CBET saturation level. Different nonlinear processes, namely, ion trapping and secondary instability FSRS and BSBS, have different intensity thresholds. Here, CBET scaling with laser intensity is examined with beam diameter 140 μm at various densities $ n e / n cr$, temperature ratios $ T e / T i$, and crossing angles θ in Fig. 5. From these results, we discuss further two distinctly different behaviors of CBET gain depending on the IAW damping rate and nonlinearities: it may grow continuously to reach a steadystate value, or it may increase to a peak value and then drop to a much lower value, especially as the IAW damping rate increases. Both behaviors can be associated with saturation by ion trapping detuning and FSRS/BSBS. We show below that CBET gain does not always increase with intensity—it may decrease with laser intensity in the presence of iontrapping induced detuning and saturation as a result of secondary instabilities.
We first discuss CBET gain behavior at $ n e / n cr = 0.04$ and $ \theta = 47 \xb0$, IAW $ k \lambda D = 0.30$, but with different temperature ratios $ T e / T i = 4$ and 2. The linear properties are the same as those discussed for Fig. 4 (for $ T e / T i = 4 , \u2009 v ph / v th , i = 3.38 , \u2009 \omega r / \omega pe = 0.005 \u2009 91 , \omega i / \omega pe = \u2212 0.000 \u2009 26$ for the resonant IAW; for $ T e / T i = 2 , \u2009 v ph / v th , i = 2.77 , \u2009 \omega r / \omega pe = 0.006 \u2009 85 , \u2009 \omega i / \omega pe = \u2212 0.000 \u2009 89$), where we showed that CBET gain reduces with temperature ratio. Figure 5 shows the CBET gain from simulations at intensities $ I ave = 7.5 \xd7 10 13$ to $ 6.0 \xd7 10 14$ W/cm^{2} for $ T e / T i = 4$ (a1) and at intensities $ I ave =$ 1.5 and $ 3.0 \xd7 10 14$ W/cm^{2} for $ T e / T i = 2$ (a2). With $ T e / T i = 4$, the IAW damping rate is smaller, the wave phase velocity is larger, and the IAW interact with the tail ions, the intensity threshold for the onset of ion trapping is lower, $ I th = 7.5 \xd7 10 13$ W/cm^{2}, and in general, CBET increases in time and saturates at an increasingly higher steadystate level as the intensity increases. With $ T e / T i = 2$, the IAW damping rate is larger, the wave phase velocity is smaller, the IAW interact with ions closer to the bulk of the distribution, and the onset of trapping intensity threshold is higher $ I th = 1.5 \xd7 10 14$ W/cm^{2}. In the latter case, CBET gain vs time can exhibit two different behaviors depending on the intensity. At higher intensity, the CBET gain increases to its peak value and then drops to a much lower steadystate value, while at lower intensity, it increases in time, reaching a steadystate value. In both temperature ratio cases, FSRS is unstable in the amplified seed beam when intensity $ I ave \u2265 3.0 \xd7 10 14$ W/cm^{2}.
We now discuss CBET gain behavior at the same crossing angle $ \theta = 47 \xb0$ but at much lower density $ n e / n cr = 0.01$ and higher IAW $ k \lambda D = 0.61$. With $ T e / T i = 4$, the linear properties are $ v ph / v th , i = 3.15 , \u2009 \omega r / \omega pe = 0.011 \u2009 18 , \u2009 \omega i / \omega pe = \u2212 0.000 \u2009 71$ for the resonant IAW. Figure 5(b) shows the CBET gain from simulations at intensities $ I ave = 7.5 \xd7 10 13$ to $ 1.2 \xd7 10 15$ W/cm^{2}. Here, the IAW damping rate is smaller than that in frame (a2), and the onset threshold intensity for ion trapping is lower $ I th = 7.5 \xd7 10 13$ W/cm^{2}. Ion trapping and its detuning effects increase with intensity, but FSRS and BSBS are absent in these simulations at the low plasma density.
Next, we consider CBET gain at density $ n e / n cr = 0.04$ but with larger crossing angle $ \theta = 99 \xb0$, resulting in IAW $ k \lambda D = 0.57$. With $ T e / T i = 4$, the kinetic linear solutions are $ v ph / v th , i = 3.18 , \u2009 \omega r / \omega pe = 0.010 \u2009 60 , \u2009 \omega i / \omega pe = \u2212 0.000 \u2009 64$ for the resonant IAW. Figure 5(c) shows the CBET gain from simulations at intensities $ I ave = 3.0$ and $ 6.0 \xd7 10 14$ W/cm^{2}. With the IAW damping further decreased compared with that in Fig. 5(b), ion trapping effects are strong. FSRS is absent (present) for the higher (lower) intensity case. At this nearly perpendicular crossing angle, FSRS is active at the higher intensity case in a subregion in the beam overlapping area close to the seed beam exiting boundary and has discernible effects on the CBET gain as indicated by the variations in the solid curve later than 8 ps. The main effect of FSRS on CBET gain is that it changes the laser beam intensity.
For $ \theta = 47 \xb0$, CBET gain behavior is examined for $ T e / T i =$ 4 and 2 at $ n e / n cr = 0.06 , \u2009 0.05 , \u2009 0.03 , \u2009 0.02$, corresponding to IAW $ k \lambda D = 0.24 , \u2009 0.27 , \u2009 0.35 , \u2009 0.43$, respectively. For $ n e / n cr = 0.06$, with $ T e / T i = 4$ ( $ T e / T i = 2$), the kinetic linear solutions are $ v ph / v th , i = 3.41$ ( $ v ph / v th , i = 2.80$), $ \omega r / \omega pe = 0.004 \u2009 81$ ( $ \omega r / \omega pe = 0.005 \u2009 58$), $ \omega i / \omega pe = \u2212 0.000 \u2009 20$ ( $ \omega i / \omega pe = \u2212 0.000 \u2009 69$). Figure 5(d1) and 5(d2) contrast the CBET gain vs intensity at the two different temperature ratios. With $ T e / T i = 4$ at both intensities $ I ave = 3.0$ and $ 6.0 \xd7 10 14$ W/cm^{2}, ion trapping effects are strong, and both FSRS and BSBS are unstable in the amplified seed beam. As a result of saturation by ion trapping detuning, FSRS and BSBS, CBET increases to its peak value and then drops to a much lower value. With $ T e / T i = 2$ and increased IAW damping rate, ion trapping effects are small for intensities $ I ave = 1.5 , 3.0$, and $ 6.0 \xd7 10 14$ W/cm^{2} with an onset threshold $ I th = 1.5 \xd7 10 14$ W/cm^{2}. FSRS in the amplified seed beam increases with intensity but at a reduced level compared with that for $ T e / T i = 4$ and BSBS is absent in the amplified seed beam. Also, with $ T e / T i = 2$ and the highest intensity, CBET increases to a peak value and then drops to a much lower value due to saturation by iontrapping detuning and FSRS, while at the lower intensities, CBET grows continuously to reach a steadystate value. With both temperature ratios, the CBET level is the lowest at the highest intensity due to stronger nonlinear saturation.
The presence of BSBS in the simulations is detected by the three methods mentioned previously: measurement of the seed beam flux on the injection boundary, examination of the FFT power spectrum of laser E_{y} along the beam injection boundary, and modification of the ion distribution function consistent with characteristics of BSBS daughter IAW. However, CBET is a seeded instability, and its IAW is active in the beam overlapping region, whereas BSBS is an instability that grows from a spectrum of noise in the simulations and its IAW may grow to large amplitude in the region near the seed beam injection boundary due to the convective nature of the instability. For the geometry shown in Fig. 1, the IAW for CBET propagates in the $ \u2212 z \u0302$ direction, while the BSBS daughter IAW propagates in the direction of the seed beam. For strong BSBS at high intensity, as in the case shown in Fig. 5(d1), the two different IAW may coexist in a small area in the beam crossing region close to the seed beam injection boundary. However, due to the very different propagation directions, the two waves modify separate regions in the velocity space distribution function. The 1D reduced distribution along the $ z \u0302$ direction around the CBET IAW phase velocity is not modified in a discernible way by the presence of BSBS daughter IAW. Thus, the waveparticle interaction for the CBET process is unaffected by the presence of the BSBS daughter IAW. The main effect of BSBS on CBET gain is that it changes the seed beam intensity.
For $ n e / n cr = 0.05$, with $ T e / T i = 4$ ( $ T e / T i = 2$), the kinetic linear solutions are $ v ph / v th , i = 3.40$ ( $ v ph / v th , i = 2.79$), $ \omega r / \omega pe = 0.005 \u2009 28$ ( $ \omega r / \omega pe = 0.006 \u2009 12$), and $ \omega i / \omega pe = \u2212 0.000 \u2009 22$ ( $ \omega i / \omega pe = \u2212 0.000 \u2009 77$). In Figs. 5(e1) and 5(e2), we compare the CBET gain at the two different temperature ratios. Similar to the above discussion on simulation results for $ n e / n cr = 0.06$, with $ T e / T i = 4$, ion trapping effects are strong; at the higher intensity, both FSRS and BSBS are unstable in the amplified seed beam and CBET increases to its peak value and then drops to a much lower value due to saturation by iontrapping detuning, FSRS, and BSBS. At the lower intensity, only FSRS is unstable, and CBET grows to a higher steadystate value. With $ T e / T i = 2$ and increased IAW damping, ion trapping effects are small with a higher onset threshold at $ I th = 3.0 \xd7 10 14$ W/cm^{2}, FSRS is at a reduced level, and BSBS is absent in the amplified seed beam. At the highest intensity, CBET increases to its peak value and then drops to a much lower value due to saturation by iontrapping detuning and FSRS, while at the lower intensities, CBET grows continuously to reach a steadystate value. With both temperature ratios, the CBET level is also the lowest at the highest intensity due to nonlinear saturation.
For $ n e / n cr = 0.03$, with $ T e / T i = 4$ ( $ T e / T i = 2$), the kinetic linear solutions are $ v ph / v th , i = 3.35$ ( $ v ph / v th , i = 2.75$), $ \omega r / \omega pe = 0.006 \u2009 79$ ( $ \omega r / \omega pe = 0.007 \u2009 89$), and $ \omega i / \omega pe = \u2212 0.000 \u2009 31$ ( $ \omega i / \omega pe = \u2212 0.001 \u2009 05$). In Figs. 5(f1) and 5(f2), the CBET gains vs intensity are shown at the two different temperature ratios. At this lower density, BSBS is absent. With $ T e / T i = 4$, ion trapping effects are strong, and CBET grows continuously to reach a steadystate value with the presence of FSRS, which is low for the lower intensity case. With $ T e / T i = 2$ and higher IAW damping, ion trapping effects are small with an onset threshold at $ I th = 1.5 \xd7 10 14$ W/cm^{2} and FSRS is present only at the highest intensity. The behavior of CBET changes from being time dependent at the highest intensity, in which saturation is by iontrappinginduced detuning and FSRS, to a continuous growth to a steadystate at the lowest intensity. With both temperature ratios, the CBET level increases with intensity similar to the case for $ n e / n cr = 0.04$ shown in Figs. 5(a1) and 5(a2).
Finally, for $ n e / n cr = 0.02$, with $ T e / T i = 4$ ( $ T e / T i = 2$), the kinetic linear solutions are $ v ph / v th , i = 3.29$ ( $ v ph / v th , i = 2.71$), $ \omega r / \omega pe = 0.008 \u2009 22$ ( $ \omega r / \omega pe = 0.009 \u2009 58$), and $ \omega i / \omega pe = \u2212 0.000 \u2009 41$ ( $ \omega i / \omega pe = \u2212 0.001 \u2009 36$). CBET gain vs intensity are shown in in Figs. 5(g1) and 5(g2). At this low density, BSBS is absent, and FSRS only occurs at a low level with $ T e / T i = 4$ at the highest intensity. With $ T e / T i = 4$, ion trapping effects are strong, and CBET grows continuously to reach a steadystate value except a slight drop with time at the higher intensity. With $ T e / T i = 2$ and higher IAW damping, ion trapping effects are small with an onset threshold at $ I th = 1.5 \xd7 10 14$ W/cm^{2}. In this case, CBET transitions from being time dependent at the highest intensity, in which saturation is mainly by iontrappinginduced detuning, to a continuous growth to a steadystate at the lowest intensity. With both temperature ratios, the CBET level increases with intensity similar to the case for $ n e / n cr = 0.04$ and 0.03.
D. Effects of density on CBET gain at a range of crossing angles
We consider CBET scaling with density for a given intensity $ I ave = 3.0 \xd7 10 14$ W/cm^{2}, beam diameter 140 μm, and temperature ratio $ T e / T i = 2$, at crossing angles $ \theta = 47 \xb0 , \u2009 31 \xb0 , \u2009 and \u2009 15.3 \xb0$. From the plasma response function, we expect the gain to decrease with density if the linear theory is valid and nonlinear effects can be neglected. However, as we will show from our simulations, nonlinear effects may lead to behavior that differs from linear theory expectations.
For crossing angle $ \theta = 47 \xb0$, the linear properties for density $ n e / n cr = 0.06 , \u2009 0.05 , \u2009 0.04 , \u2009 0.03 , \u2009 0.02$ are IAW $ k \lambda D = 0.24 , \u2009 0.27 , 0.30 , \u2009 0.35 , \u2009 0.43$, and IAW damping rates $ \omega i / \omega pe = \u2212 0.00070 , \u2212 0.00077 , \u2212 0.00089 , \u2212 0.00105 , \u2212 0.00136$, i.e., damping increases as the density decreases. The corresponding plasma linear response functions in Fig. 6(a1) show that the resonant frequency $ \omega r / \omega pe$ (i.e., the frequency corresponding the peak of the function) increases with decreasing density. Also, as density decreases, the peak value of $ Im \u2009 \Gamma ( \omega , k IAW )$ decreases, while its width (which reflects the IAW damping) increases. These quantities change monotonically with the density and CBET gain from the linear theory is expected to decrease with density.^{5} The CBET gain scaling with density from simulations, on the other hand, does not reduce with density monotonically, as shown in Fig. 6(a2) (at $ I ave = 3.0 \xd7 10 14$ W/cm^{2}). With decreasing density, the corresponding IAW phase velocity decreases (though not a sensitive function of density), and the nonlinear effects vary, as FSRS is a strong function of density (see Table I). At $ n e / n cr = 0.04$ (green curve), the CBET gain increases to a peak value and then drops to a much lower value due to saturation by iontrappinginduced detuning and FSRS.
$ n e / n cr$ .  $ v ph / v th , i$ .  Ion trapping .  FSRS level . 

0.06  2.796  Small  High 
0.05  2.787  Small  High 
0.04  2.774  Moderate  Moderate 
0.03  2.753  Small  Absent 
0.02  2.714  Small  Absent 
$ n e / n cr$ .  $ v ph / v th , i$ .  Ion trapping .  FSRS level . 

0.06  2.796  Small  High 
0.05  2.787  Small  High 
0.04  2.774  Moderate  Moderate 
0.03  2.753  Small  Absent 
0.02  2.714  Small  Absent 
For $ \theta = 31 \xb0$, the linear properties for density $ n e / n cr = 0.06 , 0.05 , \u2009 0.04 , \u2009 0.03$ are IAW $ k \lambda D = 0.16 , \u2009 0.18 , \u2009 0.20 , \u2009 and \u2009 0.23$, and the IAW damping rate $ \omega i / \omega pe = \u2212 0.000 \u2009 45 , \u2212 0.000 \u2009 50 , \u2212 0.000 \u2009 56 , and \u2212 0.000 \u2009 66$ (damping increases as the density decreases). The corresponding plasma linear response functions in Fig. 6(b1) show a monotonic increase in the resonant frequency $ \omega r / \omega pe$ as the density and the peak value of $ Im \u2009 \Gamma ( \omega , k IAW )$ decrease. Although the trend of the CBET gain scaling with density from simulations shown in Fig. 6(b2) agrees more with the linear theory than the above case, nonlinear effects are observed, as shown in Table II.
$ n e / n cr$ .  $ v ph / v th , i$ .  Ion trapping .  FSRS level . 

0.06  2.819  Strong  Moderate 
0.05  2.815  Moderate  Moderate 
0.04  2.809  Small  Moderate 
0.03  2.799  Small  Absent 
$ n e / n cr$ .  $ v ph / v th , i$ .  Ion trapping .  FSRS level . 

0.06  2.819  Strong  Moderate 
0.05  2.815  Moderate  Moderate 
0.04  2.809  Small  Moderate 
0.03  2.799  Small  Absent 
For $ \theta \xb0 = 15.3$, at $ n e / n cr = 0.06$, 0.05, 0.04, 0.03, IAW $ k \lambda D = 0.081$, 0.089, 0.100, 0.116, and $ \omega i / \omega pe = \u2212 0.000 \u2009 22$, –0.000 24, –0.000 27, and –0.000 32. The plasma linear response functions are shown in Fig. 6(c1). With decreasing density, the IAW phase velocities and FSRS levels are shown in Table III. No significant ion trapping is found in these cases. The CBET gain scaling with density from simulations is shown in Fig. 6(c2) in which the timevarying nature of the gain at the three higher densities is due to the various levels of FSRS (in the absence of ion trapping effects). There is no clear resemblance between CBET gain scaling from simulations and the scaling of $ Im \u2009 \Gamma ( \omega , k IAW )$ in Fig. 6(c1).
$ n e / n cr$ .  $ v ph / v th , i$ .  Ion trapping .  FSRS level . 

0.06  2.834  Insignificant  High 
0.05  2.833  Insignificant  Midlevel 
0.04  2.831  Insignificant  Moderate 
0.03  2.829  Insignificant  Absent 
$ n e / n cr$ .  $ v ph / v th , i$ .  Ion trapping .  FSRS level . 

0.06  2.834  Insignificant  High 
0.05  2.833  Insignificant  Midlevel 
0.04  2.831  Insignificant  Moderate 
0.03  2.829  Insignificant  Absent 
These results indicate that linear plasma response alone is unable to explain the systemintegrated CBET gain, which is determined by a combination of effects from a variety of nonlinear processes, necessitating the development of a more sophisticated model to capture this behavior.
E. Effects of crossing angles and interaction length on CBET gain
For a given beam diameter $ d = 140 \u2009 \mu $ m, the interaction length L increases with decreasing crossing angle θ as $ L = d / \u2009 sin \u2009 ( \theta )$ in the 2D simulations. For $ \theta = 15.3 \xb0 , \u2009 31.0 \xb0 , \u2009 47.0 \xb0 , \u2009 L = 541$, $ 272$, and $ 191 \u2009 \mu $ m, respectively. In this section, we explore the CBET gain scaling with θ for different density $ n e / n cr = 0.06$, 0.05, 0.04, and 0.03 at $ T e / T i = 2$ and $ I ave = 3.0 \xd7 10 14$ W/cm^{2}. As θ increases, IAW $ k \lambda D$ increases at each density value, and $ Im \u2009 \Gamma ( \omega , k IAW )$ at the resonant IAW $ k \lambda D$ is shown in Figs. 7(a1)–7(d1).
For $ n e / n cr = 0.06$, with increasing θ, IAW $ k \lambda D$ increases, $ v ph / v th , i$ decreases, IAW damping rate $ \omega i / \omega pe$ increases, and nonlinear effects vary as shown in Table IV. For the smallest $ k \lambda D$ in the absence of ion trapping, FSRS leads to saturation of the CBET gain at the lowest level even though the case has the largest peak value for $ Im \u2009 \Gamma ( \omega , k IAW )$.
θ .  $ k \lambda D$ .  $ v ph / v th , i$ .  $ \omega i / \omega pe$ .  Ion trapping .  FSRS . 

$ \theta = 15.3 \xb0$  0.08  2.834  −0.000 22  Absent  High 
$ \theta = 31.0 \xb0$  0.16  2.819  −0.000 45  Strong  Moderate 
$ \theta = 47.0 \xb0$  0.24  2.796  −0.000 69  Small  High 
θ .  $ k \lambda D$ .  $ v ph / v th , i$ .  $ \omega i / \omega pe$ .  Ion trapping .  FSRS . 

$ \theta = 15.3 \xb0$  0.08  2.834  −0.000 22  Absent  High 
$ \theta = 31.0 \xb0$  0.16  2.819  −0.000 45  Strong  Moderate 
$ \theta = 47.0 \xb0$  0.24  2.796  −0.000 69  Small  High 
For $ n e / n cr = 0.05$, with increasing θ, the linear properties and nonlinear effects are shown in Table V. Similar to the above case, for the smallest $ k \lambda D$, FSRS saturates the CBET gain to the lowest level even though the case has largest peak of $ Im \u2009 \Gamma ( \omega , k IAW )$.
θ .  $ k \lambda D$ .  $ v ph / v th , i$ .  $ \omega i / \omega pe$ .  Ion trapping .  FSRS . 

$ \theta = 15.3 \xb0$  0.09  2.833  −0.000 24  Absent  Midlevel 
$ \theta = 31.0 \xb0$  0.18  2.815  −0.000 50  Moderate  Moderate 
$ \theta = 47.0 \xb0$  0.27  2.787  −0.000 77  Small  High 
θ .  $ k \lambda D$ .  $ v ph / v th , i$ .  $ \omega i / \omega pe$ .  Ion trapping .  FSRS . 

$ \theta = 15.3 \xb0$  0.09  2.833  −0.000 24  Absent  Midlevel 
$ \theta = 31.0 \xb0$  0.18  2.815  −0.000 50  Moderate  Moderate 
$ \theta = 47.0 \xb0$  0.27  2.787  −0.000 77  Small  High 
For $ n e / n cr = 0.04$, with increasing θ, the linear properties and nonlinear effects are shown in Table VI. With the varying levels of trapping and FSRS, there is no clear relation between the CBET gain scaling from simulations and the scaling of $ Im \u2009 \Gamma ( \omega , k IAW )$.
θ .  $ k \lambda D$ .  $ v ph / v th , i$ .  $ \omega i / \omega pe$ .  Ion trapping .  FSRS . 

$ \theta = 15.3 \xb0$  0.1  2.831  −0.000 27  Absent  Moderate 
$ \theta = 31.0 \xb0$  0.2  2.809  −0.000 56  Small  Moderate 
$ \theta = 47.0 \xb0$  0.3  2.774  –0.000 89  Moderate  Moderate 
θ .  $ k \lambda D$ .  $ v ph / v th , i$ .  $ \omega i / \omega pe$ .  Ion trapping .  FSRS . 

$ \theta = 15.3 \xb0$  0.1  2.831  −0.000 27  Absent  Moderate 
$ \theta = 31.0 \xb0$  0.2  2.809  −0.000 56  Small  Moderate 
$ \theta = 47.0 \xb0$  0.3  2.774  –0.000 89  Moderate  Moderate 
For $ n e / n cr = 0.03$, with increasing θ, the linear properties and nonlinear effects are shown in Table VII. In the absence of FSRS and strong ion trapping effects, here the trend of the CBET gain scaling with θ from simulations agrees with that for the scaling of $ Im \u2009 \Gamma ( \omega , k IAW )$.
θ .  $ k \lambda D$ .  $ v ph / v th , i$ .  $ \omega i / \omega pe$ .  Ion trapping .  FSRS . 

$ \theta = 15.3 \xb0$  0.12  2.829  −0.000 32  Absent  Absent 
$ \theta = 31.0 \xb0$  0.23  2.799  −0.000 66  Small  Absent 
$ \theta = 47.0 \xb0$  0.35  2.753  −0.001 05  Small  Absent 
θ .  $ k \lambda D$ .  $ v ph / v th , i$ .  $ \omega i / \omega pe$ .  Ion trapping .  FSRS . 

$ \theta = 15.3 \xb0$  0.12  2.829  −0.000 32  Absent  Absent 
$ \theta = 31.0 \xb0$  0.23  2.799  −0.000 66  Small  Absent 
$ \theta = 47.0 \xb0$  0.35  2.753  −0.001 05  Small  Absent 
The smallest θ corresponds to the longest interaction length L and smallest IAW damping rate. However, the results in this section show that in the presence of various level of nonlinear trapping and secondary instability FSRS, CBET gain does not necessarily increase with the interaction length.
F. Effects of beam diameter and interaction length on CBET gain
The effects of increasing interaction length on CBET gain from increasing the beam diameter are also examined for different density, crossing angle, and IAW $ k \lambda D$ values. In particular, we explore whether CBET gain saturates or continues to increase as the beam diameter increases. Figure 8(a) shows CBET gains for three laser beam diameters 140 μm (solid), 280 μm (dashed), and 420 μm (dotted) for a parameter space where ion trapping is present in all three simulations; however, FSRS varies (see caption for parameters). With the increasing beam diameter and interaction length, FSRS in the amplified seed beam and pump depletion increase, limiting the CBET gain in the case with the largest diameter beams.
Similarly, Fig. 8(b) shows CBET gains for the three laser beam diameters in the absence of FSRS in the amplified seed in all cases. The ion trapping is strong, and the CBET gain continues to increase with the beam diameter.
Additionally, Fig. 8(c) shows CBET gains for the three beam diameters at near perpendicular crossing angle $ \theta = 99 \xb0$. The CBET gain scaling with beam diameter is similar to that in Fig. 8(a), except the gain becomes smaller at the largest beam diameter 420 μm than at the intermediate diameter 280 μm. Ion trapping is present in all three cases, and FSRS in the amplified seed beam changes from being absent for the smallest diameter beam to increasing with the beam diameter for the two larger beam diameter cases, limiting the CBET gain level at the largest diameter case.
V. CBET SCALING FIGURE OF MERIT AND NONLINEAR MODEL
The threshold intensity for the onset of the ion trapping is different ranging from $ I th = 7.5 \xd7 10 13$ to $ 3.0 \xd7 10 14$ W/cm^{2} depending on the IAW damping as discussed in Sec. IV C.
Linear CBET models are inadequate when nonlinear effects are important. To see this, in Fig. 9(e), we have compared a subset of VPIC simulations and the nonlinear CBET model with the linear theory predictions using a form in the appendix of Ref. 25 [Eqs. (A8) and (A9), i.e., based on Ref. 60.] The parameters and VPIC simulations for the comparison are from Fig. 5(a1), and the timeaveraged VPIC gain % defined in Sec. II is converted to gain $ G = ln \u2009 ( P s \u200a out / P s \u200a in )$ as used in the linear theory. Values of gain G from the linear theory (red) and VPIC simulations (black) shown in the inset clearly demonstrate the CBET nonlinear saturation captured in the VPIC simulations and the nonlinear CBET model, which is absent in linear description. As a consequence, while agreement is observed at low intensity, the linear theory predictions (red open circles) differ significantly from VPIC simulations (red solid disks) and the nonlinear CBET model predictions at high intensity for this density range. (Note that this form of the linear CBET gain does not depend on the seed beam intensity and has no saturation at high intensity. For the same pump and seed intensity, the linear theory yields the unphysical result that the seed beam gains more energy than is available in the pump beam.)
For modeling the plasma heating associated with CBET, we use the Manley–Rowe relations as a reference heating power, recognizing that they are not a rigorous description of the underlying physics owing to complex ion trapping and secondary instability effects in the simulations. To account for nonideal effects, we use a multiplier on the M–R heating power. The physics assumptions giving rise to the Manley–Rowe relations are that the system evolves to a timeasymptotic partitioning of energy among three coupled oscillators of fixed frequencies; these assumptions may not be satisfied in the case of CBET where, e.g., secondary instabilities may arise. Even when secondary instabilities (BSBS or FSRS) are not present, time and spatialdependent ionwavefront breakup may occur, introducing transverse modes in the system, and convection can propagate plasma wave energy outside the interaction region. Both will break the Manley–Rowe assumptions, leading to the need for a multiplier on the M–R heating power.
Using the CBET gain from simulations and Eq. (5), we can compute $ P IAW MR$ empirically. On the other hand, power into ions $ P ion$ via Landau damping of IAWs can be obtained from time profiles of energy for the ions in the simulation box and those leaving the simulation boundary. For the purposes of generating our model, we have calculated the power into the ions both in the system and exiting the simulation boundaries and subtracting the measured thermal fluctuations of the plasma from a second simulation without laser drivers.
However, when we compare power into ions via Landau damping of IAWs $ P ion$ to $ P IAW MR$ computed using Eq. (5), we find that they may agree or differ significantly depending on the intensity and various levels of nonlinear effects present in the simulations. We illustrate this using the results from two parameter settings at different IAW damping rates. The first setting is with lower IAW damping for IAW $ k \lambda D = 0.30$ shown in Fig. 10(a). The solid curves are power in ions $ P ion$ vs laser intensity, while the dotted curves are $ P IAW MR$ computed empirically using Eq. (5) (see parameters in the caption). At lower IAW damping, ion trapping can enhance the power into the ions, and we find that $ P ion / P IAW MR \u223c 1$ at lower laser intensity but $ P ion / P IAW MR > 1$ at higher intensities with its value increasing with intensity. Note that FSRS are present in the amplified seed beam at higher laser intensity, and FSRS daughter EPW can decay into a counterpropagating EPW and an IAW (i.e., Langmuir decay instability, or LDI^{61}) contributing to the increase in the power in the ions.
The second parameter setting is with higher IAW damping at IAW $ k \lambda D = 0.61$ in Fig. 10(b) at low density $ n e / n cr = 0.01$ where FSRS and BSBS are absent and the only nonlinear effects are ion trapping (in addition to possible pump depletion). At increased IAW damping, iontrappinginduced detuning effects increase with intensity and we find that $ P ion / P IAW MR \u223c 1$ at lower laser intensity but $ P ion / P IAW MR < 1$ at higher intensities with its value decreasing with intensity.
Using VPIC simulations at intensities at and above the onset threshold for ion trapping, we have constructed nonlinear CBET models for the energy transfer and for ion heating, expressed as a function of key laser and plasma conditions shown in Eqs. (2)–(6). These results can be used in laser raytracing package together with a linear CBET model. Along a ray path, when laser intensity is above the onset laser intensity for ion trapping, the linear CBET model can be replaced by the nonlinear model to compute the power change per length along the ray as well as the energy deposition into plasma derived from the nonlinear expressions. The nonlinear CBET model implementation in raytracing package should be constrained by the saturation effects shown in Fig. 8 such that the seed beam power gain does not accumulate to reach values larger than the predicted saturation level after integration over longer path length. Initial implementation of this approach in the Rage radhydro code has been explored^{62} and is showing encouraging results.
While more data are needed to improve statistics of the nonlinear CBET model, the current work has laid out an approach for constructing such a model using key laser and plasma parameters. It is apparent that the need for five key parameters I_{14}, $ k \lambda D$, L, $ n e / n cr$, and $ Z T e / T i$ requires a large dataset for reasonable statistical representation in models. From the physics bases for these key parameters as presented in Secs. III and IV, it is unclear any of these key parameters should be omitted, as VPIC simulations presented in this work have shown their significant effects on CBET gain. The approach using closed form expressions presented here has the advantage of being relatively straightforward to implement in design codes. In future work, we will also explore the use of a subset of VPIC simulation data to construct nonlinear CBET models targeting applications to specific ICF facilities and/or platforms based on their laser and plasma conditions. This may reduce the number of key parameters in the nonlinear CBET model.
Another approach (which we will report in a future manuscript) would be to use these results of our VPIC simulations as training data in the generation of a Gaussian process emulator.^{63} This approach has the benefit of being straightforward to update as additional simulation data becomes available.
VI. OTHER EFFECTS ON CBET DYNAMICS NOT INCLUDED IN THE NONLINEAR CBET MODEL
A. Effects of collisions on the CBET gain
Although the majority of the simulations in this work are collisionless, some simulations are considered here that include the effects of binary collisions for cases with varying levels of IAW damping, nonlinear effects, CBET gain, and different timedependent characteristics of gain. The simulations were performed using a binary collision model^{34} and include all self and crossspecies collisions. To conserve energy and momentum rigorously in the collision operator, all computational electron and ion macroparticles are chosen to have the same statistical weights, meaning that in a given simulation, each particle, whether electron or ion, represents the same number of physical electrons or ions. The collision operator uses a fixed Coulomb logarithm $ log \u2009 \Lambda $^{64} for all collisions (electron–electron and electron–He^{2+} collisions). The collision model has been used in prior LPI studies^{25,65} of Trident shortpulse experiments,^{66,67} Jupiter experiments,^{26} and recent CBET experiments,^{31,32} as well as in other settings, including magnetic reconnection^{68} and interfacial mix.^{69,70}
Since Coulomb collision rates scale dominantly as $ Z i 2 Z j 2 T e \u2212 3 / 2$ where Z_{i} and Z_{j} are the charges of the particles, collisions are important in the modeling of lowertemperature LPI experiments or in plasma media with midZ or highZ components.^{25,31,32,71,72} In our present study of plasma with He^{2+} ions, collisional effects are not expected to play a significant role at a density of a few percent $ n cr$ with keV temperature. For example, with T_{e} = 3 keV, $ n e / n cr = 0.04 , \u2009 \u2009 log \u2009 \Lambda = 8.3$, the number of e–e, He–He, and He–e collisions per ps is small, 0.20, 0.15, and 0.07, respectively. For the collisional effects on resonant He ions, He–e scattering is small compared to He–He scattering. For He–He collisions, the perpendicular scattering rate is comparable to the slowingdown rate, and both are much larger than the parallel scattering rate. The estimated timescale for collisions is greater than 100 ps, longer than a typical simulation time. As a result, the ion distributions in the simulations show persistent trapping modifications to the distribution functions.
In collisional simulations, electrons may experience inversebremsstrahlung heating and, as a result, the temperature ratio $ T e / T i$ and the IAW damping rate may evolve.^{30} Effects of collisions also include detrapping of ions in IAW and electrons in EPW when FSRS is active. Thus, collisions can reduce or enhance the CBET gain, depending on the level of ion and electron trapping and how far the laser intensity is above the threshold for ion trapping and secondary instabilities. For cases at intensity just above the onset of ion trapping, collisional detrapping of ions can weaken the trapping enhancement of CBET. On the other hand, collisions can increase the CBET level if collisions weaken the trappinginduced detuning of CBET. Collisions can also increase the onset threshold intensity for FSRS and reduce the FSRS level. We found that collisional effects on CBET gain are, in general, insignificant in indirectdrive settings, unlike the situation found in lower temperature plasma and in the presence of midZ ions.^{25}
Figure 11(a) shows the CBET gain with (dotted curve) and without (solid) collisions from a case with small IAW damping, strong ion trapping, and the presence of FSRS in the amplified seed beam (see parameters in figure caption). With and without collisions, ion trapping effects and secondary instability FSRS levels are comparable, and the CBET gain is similar with a 6% decrease in the timeaveraged gain value for the collisional case. The collisional heating on the electrons increases electron temperature T_{e} by a few percent.
Figure 11(b) is for a lower CBET gain case and with parameters similar to (a) but with larger IAW damping at a lower $ T e / T i$, a smaller wave phase velocity $ v ph / v th , i$, and a lower intensity. Both ion trapping and FSRS are moderate without collisions but they are small with collisions. These results indicate that reduction of ion trapping and FSRS by collisions can enhance the CBET gain. The CBET gains for these two simulations are similar with less than 2% increase in the timeaveraged gain value for the collisional case.
Figure 11(c) is for a case with strong ion trapping but without FSRS. The CBET gains are similar with less than a 10% increase in the timeaveraged gain value for the collisional case. Figure 11(d) is for a case with smaller $ v ph / v th , i$, small ion trapping, and no secondary instabilities. The CBET gains are similar with less than a 3% increase in the timeaveraged gain value for the collisional case.
B. Effects of plasma flow on CBET gain
In the presence of plasma flow, CBET can also occur between monochromatic beams.^{73} For simulations with plasma flow, the pump and seed beam frequencies are the same and the phase speed of the IAW $ v ph = \u2212 k \xb7 V / k$ where $V$ is the plasma flow velocity and $k$ is the resonant IAW wavenumber determined from the light wavenumber and the crossing angle.
For CBET simulations with spatially uniform flow velocities, we load the simulation particles with a uniform density, temperature, and Maxwellian velocity distributions with a constant drift. We employ open particle boundary conditions with plasma flow by assuming that the distribution functions of the plasma species just outside the simulation volume are described by drifting Maxwellians with density and temperature the same as the initial conditions and we inject particles in such a way that the inward fluxes of particles and their distributions are consistent with this assumption. For the laser intensities and plasma temperatures used in our study, it is sufficient to assume nonrelativistic distributions and to ignore the quiver velocity of the electrons in the laser field in the injection. The details of the generalized open boundary method are described in Ref. 75.
Here, we compare CBET simulations without flow (shown by solid curves) to their counterparts with flow (dotted curves) for cases where we observe strong to midlevel ion trapping and FSRS effects in Figs. 12(a) and 12(b), respectively (see parameters in the figure caption). No remarkable changes on ion trapping and FSRS levels are found, and the CBET gains are similar with and without flow, with less than 5% and 8% decrease in the timeaveraged gain value for the flow case for (a) and (b), respectively.
C. Effects of initial frequency mismatch on CBET gain
In our simulations, the frequency difference of the pump and seed beams, $ \omega = \omega p \u2212 \omega s$, is set to be the frequency of the IAW at the resonant IAW wavenumber. In this section, we examine the effects of a frequency mismatch on the CBET gain for $ \theta = 47 \xb0$ and $ n e / n cr = 0.04$. With higher (lower) electrontoion temperature ratio $ T e / T i = 4$ ( $ T e / T i = 2$), the IAW damping rate is lower (higher), and the plasma response function has a narrower (broader) width, as shown in Fig. 4. Here, we show the effects of a frequency mismatch on the CBET gain for the case with $ T e / T i = 4$ in which the change in the plasma response function off resonance is larger compared with that where $ T e / T i = 2$ for a given frequency mismatch. In Fig. 13, the imaginary part of the plasma response function $ Im \u2009 \Gamma ( \omega , k IAW )$ at resonant IAW wavenumber and the CBET gain from simulation with resonant matching conditions is shown by the solid curves where $ \omega p \u2212 \omega s = 0.0059 \omega pe$ (see parameters in the figure caption). We compared this with a case in which $ \omega p \u2212 \omega s = 0.0055 \omega pe$, downshifted from the matching frequency by 7% as indicated by the dotted line in (a). From the linear theory, the plasma response function reduces by nearly 60%. However, the CBET gain shown by the dotted curve in (b) is not significantly affected by the frequency mismatch, with less than a 6% decrease in the timeaveraged value. In this case, due to the dominant effects of the iontrapping induced frequency shift in simulations, the nonlinear CBET gain appears to be insensitive to the amount of initial frequency mismatch.
D. Effects of nonMaxwellian electrons on the CBET gain
While the simulations in this work use initial Maxwellian distributions for electrons and ions, we examine possible effects of nonMaxwellian electrons on the CBET gain. It is known that inverse bremsstrahlung heating from laser can lead to a superGaussian electron distributions when the ee equilibration time is long compared to the heating time.^{75,76} In such cases, a nonMaxwellian electron distribution may affect the CBET gain under conditions relevant to ICF.^{15} Here, we use a superGaussian of order 3 as measured in prior Omega CBET experiments^{31} to assess its effects on the CBET gain in the parameter space for this work.
Figure 14 shows an example case comparing initial Maxwellian vs nonMaxwellian electron distributions. To isolate the effects on IAW, the case we consider has ion trapping, but FSRS is absent at a low plasma density (see parameters in the caption). As the electron distribution function changes from Maxwellian to nonMaxwellian, the resonant IAW frequency increases while damping decreases. The peak value of $ Im \u2009 \Gamma ( \omega , k IAW )$ increases and shifts to a higher frequency. However, the amount of frequency upshift is comparable to but smaller than the iontrapping induced frequency downshift shown in Fig. 2(c). As a consequence, the CBET gain in the two cases are comparable, with the timeaveraged gain decreasing by 7% in the case with nonMaxwellian electrons, indicating the dominant nonlinear effects of ion trapping on the CBET gain in this case.
VII. SUMMARY AND CONCLUSIONS
We have examined the nonlinear dynamics and saturation of CBET using largescale 2D VPIC simulations for a range of plasma and laser conditions, including various plasma densities, temperatures, laser beam intensities, crossing angles, and beam diameters. We have developed a new automated δfGaussianmixture algorithm for an accurate characterization of trappinginduced nonMaxwellian distributions and the resulting plasma response. We have shown that modification of the ion distribution function resulting from trapping leads to a reduction of IAW damping and an enhancement of the gain, followed by a shift of the peak of the plasma response function toward lower frequency and a reduction of its value at the driving frequency, thus detuning CBET.
We have discussed the nonlinear effects on the timedependent growth and saturation of CBET resulting from ion trapping and secondary FSRS and BSBS instabilities, which exhibit different onset intensity thresholds. The iontrappinginduced detuning, IAW nonlinearities, speckle interaction, and CBET gain in local regions depend on the in situ, timedependent speckle intensities in the beam crossing region resulting from pump beam depletion, seed beam amplification, and levels of FSRS and BSBS. FSRS in the amplified seed beam is a strong function of density and is negligible for low density plasma (e.g., $ n e / n cr \u223c 0.01$). The amount of BSBS in the amplified seed beam increases with density, temperature ratio $ T e / T i$, and laser intensity. Depending on the IAW damping rate and the levels of various nonlinear processes, the overall CBET gain can either continuously rise to a peak steadystate value or increase to a peak and then drop to a quasisteadystate value. The latter behavior is often associated with cases with larger IAW damping rate. Examples are shown by the dashed curves in Figs. 3(b) and 4(b). Due to the large angle between the two IAWs for CBET and BSBS, the waveparticle interaction for the CBET process is unaffected by the presence of the BSBS daughter IAW. The main effect of BSBS on CBET gain is that it changes the seed beam intensity. While the systemintegrated gain has reached a steadystate value, the iontrapping modification of the distribution, secondary instability activities, and CBET gain in local regions may continue to fluctuate and evolve in time (“quasisteadystate” may be a more appropriate description for the overall system).
Nonlinear effects may lead to a deviation of CBET gain from linear theory predictions in complicated ways. For instance, CBET gain does not always increase with intensity—in fact, it may decrease with laser intensity in the presence of nonlinear effects. Also, for a given laser intensity, beam diameter and crossing angle, and electrontoion temperature ratio, the scaling of CBET gain with density may behave differently from linear theory predictions that gain decreases with density. Furthermore, CBET gain does not necessarily increase with the interaction length in the presence of nonlinear trapping and secondary FSRS instability. We have also shown that the CBET gain can saturate as the laser beam diameter increases (in the presence of FSRS), or it may continue to increase as the beam diameter increases (in the absence of FSRS).
To capture these various levels of nonlinear effects on the systemintegrated CBET gain at intensities at and above the onset threshold for ion trapping, and with the ultimate goal of providing LPI CBET closures for ICF multiphysics codes in mind, we have constructed nonlinear models from VPIC simulation results for the CBET gain and the resulting energy deposition into the plasma. The models are functions of local laser and plasma conditions, namely, IAW $ k \lambda D$, temperature ratio $ T e / T i$, average pump and seed beam intensity $ I ave$, density $ n e / n cr$, beam crossing angle θ, and beam diameter d. The threshold intensity for the onset of the ion trapping is found to be ranging from $ I th = 7.5 \xd7 10 13$ to $ 3.0 \xd7 10 14$ W/cm^{2} depending on the IAW damping. In the presence of secondary FSRS and BSBS instabilities in the amplified seed beam, which are strong functions of density, improvements on the nonlinear CBET gain model can be made by dividing the density into subgroups, each corresponding to a different form of model fit. However, more simulation data are needed to improve confidence in scaling laws.
We have shown in our study that these complex ion trapping and secondary instability effects are not expected to be captured by the standard Manley–Rowe relations that consider only simple energy exchange among three coupled oscillators. The power into ions can be above or below the estimated values using the standard Manley–Rowe relation for low or high IAW damping rates. However, we can obtain an empirical model of the power into the ions from Landau damping of IAW. We do this by constructing a modified Manley–Rowe relation, which computes the classical Manley–Rowe relation using the nonlinear CBET gain model with a multiplier that depends on local laser and plasma conditions as described by Eqs. (5) and (6).
Additionally, we examined effects of collisions, plasma flow, driving frequency mismatch, and nonMaxwellian electron distribution on systemintegrated CBET gain. In our present study of plasma with He^{2+} ions for a density of a few percent $ n cr$ with keV temperature, collisional effects on systemintegrated CBET gain are found to be small in general. For collisional simulations with small IAW damping, ion trapping persists and FSRS level is comparable to that without collisions. In collisional simulations with larger IAW damping, ion trapping and FSRS are reduced, leading to a small increase in the CBET gain compared to that without collisions. For a small set of simulations with monochromatic laser bean beams and using plasma flow to satisfy the CBET resonant conditions, we found that ion trapping, FSRS, and systemintegrated gain are similar to those in simulations without flow. The effects of a frequency mismatch on the CBET gain have been examined for a lowdamping case, and the systemintegrated CBET gain was found to be insensitive to the frequency mismatch due to the dominant effects of the iontrapping induced frequency shift in simulation. Finally, the systemintegrated CBET gains with nonMaxwellian and Maxwellian electron distributions are comparable also due to the dominant effects of ion trapping on the CBET gain.
The presence of a density gradient can reduce the growth of FSRS,^{29,30} thus modifying the nonlinear evolution of CBET. Future work will include the effects of density gradients relevant to indirect drive settings to suppress FSRS and refine the nonlinear CBET models.
Finally, we comment on the effects of laser speckle geometry, dimensionality, and PIC noise on CBET and LPI in general, since these are commonly related to the physical process of sideloss of trapped particles when the laser intensity is above the onset threshold for nonlinear trapping (and other IAW nonlinearity and secondary instabilities). We use an analogy with SRS nonlinear processes resulting from particle trapping since particle trapping and FSRS are also relevant process in our CBET modeling. For a speckled laser beam, the onset threshold intensity for particle trapping is determined by collisional detrapping, sideloss of trapped particles due to the narrow speckle width, and particle noise levels in PIC simulations. In collisionless 1D PIC simulations without speckle structure, there is no sideloss, so the noise level in PIC simulations, related to the number of particles per cell (nppc) used in the simulations, determines the onset threshold intensity $ I th$ for trapping. As nppc increases, the $ I th$ decreases slowly.^{39} In 2D and 3D for speckled laser beam in the absence of collisions, given sufficient nppc, $ I th$ is determined by the sideloss of trapped particles. We showed in Sec. VI that collisional effects are not significant for the He plasma examined in this work. This is the underlying reason why SRS (or FSRS) results in nonlinear regimes where particle trapping dominate the dynamics are not sensitive to nppc. While the $ I th$ values obtained from 2D VPIC could still differ from those from 3D VPIC, the saturation level of SRS^{41} and CBET^{88} has been shown to be comparable. Thus, in regimes above the onset threshold intensity, we do not expect 3D effects to significantly change the CBET saturation levels obtained from 2D VPIC simulations. The onset threshold intensity from 2D simulations could be somewhat lower than the physical system and may need to be increased to agree with future experiments or largescale 3D PIC simulations of CBET (assuming such simulations become feasible with advances in computing).
ACKNOWLEDGMENTS
This work was performed under the auspices of the U.S. Dept. of Energy by the Triad National Security, LLC Los Alamos National Laboratory (LANL) and was supported by the LANL Directed Research and Development Program (No. 20210063DR). The authors acknowledge useful discussions with Dr. Brian Albright. VPIC simulations were run on the LANL Institutional Computing Clusters.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Lin Yin: Conceptualization (lead); Data curation (lead); Funding acquisition (equal); Investigation (lead); Methodology (lead); Project administration (lead); Writing – original draft (lead); Writing – review & editing (lead). Truong Ba Nguyen: Funding acquisition (supporting); Investigation (supporting); Methodology (equal); Software (equal); Writing – review & editing (supporting). Guangye Chen: Funding acquisition (supporting); Investigation (supporting); Methodology (equal); Software (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Luis Chacon: Conceptualization (supporting); Funding acquisition (lead); Investigation (supporting); Methodology (equal); Project administration (equal); Software (equal); Writing – original draft (supporting); Writing – review & editing (supporting). David J. Stark: Funding acquisition (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Lauren M. Green: Funding acquisition (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Brian Michael Haines: Conceptualization (supporting); Funding acquisition (equal); Project administration (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.
APPENDIX: GAUSSIANMIXTURE RECONSTRUCTION OF VPIC DATA
In our context, we seek to use GMM to reconstruct fine features developed at PDF tails due to nonlinear saturation of LPI phenomena. This turns out to be a hard problem for adaptive GMM, as it is difficult to distinguish small features in tails from noisy particle data. To overcome this problem in our context, we have developed an algorithm (which we term GMδf) that successfully identifies tail features with minor modifications to the original adaptive GMM algorithm.
The GMδf algorithm is applicable to particles directly as long as the associated particle phasespace volumes are known. In our specific application, we employ a uniform velocityspace mesh as a way to both control the total number of observations (beneficial if there are many particles per mesh point), as well as to simplify PDF resampling. For this, we begin by mapping VPIC particle data to a uniform mesh in velocity space to construct a histogram. The histogram cells will become the new “particles” (of readily available velocityspace volumes) that will drive the GMM procedure. These particles will have unequal weights, but the EM algorithm has been generalized for this case.^{87}
The GMδf algorithm has three steps, aimed at capturing small tail features: (1) a δfiteration step, (2) a component removal step (based on a desired final number of components), and (3) a final EM fitting step (to ensure fitting optimality with the selected number of components). We outline these steps in 1D for simplicity in the following. (Note the method can be straightforwardly extended to 2D.)

δf iteration step
 Approximate f(v) with an adaptive GM reconstruction step, to find$ f ( v ) \u2248 f \u0302 ( v ) = \u2211 k = 1 K \omega k G k ( v ; \mu k , \Sigma k ) .$
For this, we use the scheme proposed in Refs. 86 and 87, with a typical convergence tolerance for the maximum likelihood criterion of $ 10 \u2212 10$ or lower.

Compute $ \delta f = f ( v ) \u2212 \alpha f \u0302 ( v )$ on the velocityspace mesh with $ \alpha \u2a85 1.0$, and zero out any resulting negative values. In practice, we choose $ \alpha = 0.99$.

Reconstruct the resulting $ \delta f ( v )$ with a new adaptive GM reconstruction step. Call it $ \delta f \u0302 ( v ) .$

Compute $ \delta 2 f ( v ) = \delta f ( v ) \u2212 \alpha \delta f \u0302 ( v )$, again zeroing out negative values.

Continue the GMM reconstruction procedure n times until the relative amplitudes of Gaussian components in $ \delta n f$ with respect to the total density are small enough (typically below $ 10 \u2212 8$).

Collect all Gaussian components obtained so far.


Component removal step: depending on the number of δf iterations, the first step may result in a relatively large number of Gaussian components. The removal step begins with the user specification of the desired number of components (or a minimum ratio of Gaussian amplitudes; see below). This can be done for the whole domain, or on prescribed velocityspace subdomains (which could be selected a priori by physics intuition). Then, for each component of the mixture:

Compute $ r k = \omega k G k ( \mu k ; \mu k , \Sigma k ) \omega \u2009 G ( \mu k ; \mu , \Sigma )$ where G_{k} is the kth Gaussian, parametrized by $ ( \mu k , \Sigma k )$, and G is the largestweight Gaussian, parametrized by $ ( \mu , \Sigma )$.

If applicable, determine the subdomain that G_{k} belongs to relative to the largestweight Gaussian by computing $ ( \mu k \u2212 \mu )$.

Rank components according to r_{k} in each subdomain, and keep the largest r_{k} components following either the specified number of components, or the specified minimum value for r_{k}.


Final EMfitting step: use selected components as the initial condition for a final (nonadaptive) EM iteration to obtain the final estimate of the original PDF, f(v). We monitor covariancematrix eigenvalues during the procedure and remove components with sufficiently small eigenvalues (typically below $ 10 \u2212 5$; this is a sign of Gaussian overfitting to a single data point, a wellknown failure mode of the standard EM algorithm).
A sample GMM fit to a 1D PDF from a nonlinear CBET simulation with VPIC is shown in Fig. 15, demonstrating the ability of the scheme to match smallamplitude features on the tail with high precision.