The influences of ion trapping and fluctuations of electron temperature and plasma flow on crossbeam energy transfer (CBET) are examined using two and threedimensional particleincell simulations in parameter regimes relevant to recent CBET experiments at the OMEGA laser facility. In midZ plasma irradiated by an intense pump beam and weaker probe beam, ion trapping, collisional detrapping, and plasma flow induced by thermal effects are shown to affect the CBET gain. Ion trapping can enhance or detune the CBET resonance [Nguyen et al., Phys. Plasmas 28, 082705 (2021)]. Collisional detrapping can affect the CBET gain at low seed beam intensity near the onset threshold for ion trapping. Thermaleffectsinduced flow can also detune the CBET resonance at a level comparable to that from trapping at low seed beam intensity. As a consequence, the CBET gain is sensitive to collisions and dimensionality at low seed beam intensity where ion trapping is weak but is insensitive to collisions and dimensionality at high seed beam intensity where ion trapping is strong.
I. INTRODUCTION
Laserplasma instabilities (LPI) reduce the energy coupling to plasma and can affect both the direct and indirectdrive approaches to laserdriven initial confinement fusion (ICF).^{1–6} Two major LPI processes are stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS), where the laser light scatters off an electronplasma wave (EPW) and an ionacoustic wave (IAW), respectively.^{7} The crossbeam energy transfer (CBET), a special case of SBS, allows crossing laser beams to exchange energy through the excitation of IAW. An understanding of LPI dynamics and saturation in a variety of parameter regimes would enable the coupling of nonlinear LPI effects into ICF implosion modeling using multiphysics codes, improving their predictive capability.
To achieve this goal, dedicated LPI experimental platforms are valuable since they provide insight into LPI behavior and allow us to validate our LPI modeling approaches. Historically, these platforms have enabled an understanding of the essential nonlinear behavior of LPI in isolated laser speckles, how laser speckles interact with one another, and LPI processes in more complex settings.^{8–18}
Recent CBET experiments at the OMEGA facility^{19,20} at the Laboratory for Laser Energetics (LLE) were designed to provide a simplified, welldiagnosed CBET platform to study the interaction of a solitary seed laser beam with up to four pump beams. By isolating the CBET interaction, this platform enables detailed comparison with fully selfconsistent, nonlinear, particleincell (PIC) modeling. A recent study of this type led to an improved understanding of nonlinear CBET phenomena.^{21} Using twodimensional (2D) VPIC^{22–25} simulations, Nguyen et al. found that CBET can saturate through ion trappinginduced detuning, IAW nonlinearity, and collisional ion heating. In particular, ion trapping and heating were found to result in the peak of the plasma response function shifting to offresonant frequencies, which reduced the CBET gain. More broadly, ion trapping was found to be one of the key CBET saturation mechanisms in indirect drive settings as well.^{26–29}
Another nonlinearity that can affect CBET is laserbeam selffocusing. One form of selffocusing is ponderomotive selffocusing, arising when plasma density depressions form in highintensity beams as a result of plasma hydrodynamics response to the ponderomotive pressure.^{7} The other is thermal selffocusing from plasma heating by the inverse bremsstrahlung.^{30–37} CBET in the presence of speckledlaser beam selffocusing has been examined in studies using a fluid model of the plasma.^{38,39} It was found that ponderomotive selffocusing in speckled beams has a significant impact on CBET resulting from beam bending and plasmainduced smoothing. However, the hydrodynamic equations employed an isothermal approximation and, thus, thermal effects^{37} in laser selffocusing were not included. In Ref. 21, ion–ion collisions were used to examine ion trapping and heating with a high intensity seed beam, but thermal effects were not included since the simulations were performed without electron collisions and, hence, the inverse bremsstrahlung process. Similarly, VPIC simulations of CBET driven by largescale speckled beams^{28,29} captured ponderomotive effects but the parameters were relevant to indirect drive He^{+2} plasmas, where collisional effects and, thus, thermal effects are unimportant.
To date, the influence of ion trapping and thermal effects driven by the laser beam on CBET dynamics have not been modeled together. In this paper, we explore the nonlinear dynamics and saturation of CBET using two and threedimensional (3D) VPIC simulations of a seed beam interacting with a pump beam in parameter settings relevant to the OMEGA CBET experiments, namely, in a midZ plasma with an intense pump laser beam. To the best of our knowledge, these are the first ever multispeckled 3D fully kinetic simulations of CBET. We focus here on the influences of ion trapping and laserinduced thermal effects. In Sec. II, we describe the PIC simulation setup for modeling the OMEGA CBET experiments, including the methods for modeling the superGaussian electron distribution, interparticle collisions, and thermal effects. In Sec. III, the effects of ion trapping, collisional detrapping, and plasma flow induced by thermal effects are discussed. Results at low seed beam intensity near the onset threshold for ion trapping are contrasted with those at high seed beam intensity with strong ion trapping. Sensitivity of simulation results to collisions and dimensionality are examined in detail by comparing collisionless, only ion–ion collision and fullcollisions simulations in 2D and 3D. The main results from this work are summarized in Sec. IV.
II. MODELING OF OMEGA CBET EXPERIMENTS WITH PIC SIMULATIONS
A. Simulation setup
The plasma and laser conditions for the VPIC simulations were chosen to correspond to those of recent CBET experiments^{19,20} using the Tunable OMEGA Port9 laser.^{40} The platform employs a gasjet target comprising 45% nitrogen and 55% hydrogen in a plume heated by UV laser beams. To avoid overlapping the heater and pump/probe beams, the pump beams (with characteristic overlapping intensity $ \u223c 3.2 \xd7 10 15$ W/cm^{2}) and probe beam (with intensities ranging from 0.1 to $ 4.1 \xd7 10 14$ W/cm^{2}) were turned on 800 ps into the experiment. These beams had wavelengths of 351.11 and 351.40 nm and crossed at an angle $ 99 \xb0$ over an interaction length of 220 μm. The pump and probe beams were linearly copolarized. Using Thomsonscattering measurements, the plasma was observed to be spatially uniform over a 1.5 mm diameter plateau at initial temperatures T_{e} = 450 eV, $ T i = T e / 4$, and density $ n e = 8 \xd7 10 19$ cm^{−3}, though the density was observed to decrease slowly to $ 5.5 \xd7 10 19$ cm^{−3} over a period of 700 ps.
In the VPIC simulations, the plasma has electrons and preionized hydrogen and nitrogen ion species H^{+} and N^{7+} (with realistic mass ratios) at 55% and 45% number density, respectively, with initial Maxwellian ion and nonMaxwellian electron (superGaussian of order three) distributions, matching the experiments.^{19,20} We model the experiments with a spatially uniform, representative density $ n e = 6 \xd7 10 19$ cm^{−3} with $ n e / n cr = 0.006$ (where $ n cr$ is the critical density) and electrontoion temperature ratio $ T e / T i = 4$ and T_{e} = 600 eV. 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 for the ions and a superGaussian of order three for the electrons at initial temperatures T_{i} and T_{e}.
The simulations are performed in two spatial dimensions (2D) in the (x, z) plane and in three spatial dimensions (3D) with cell sizes nearly equal to the Debye length λ_{D} (0.02 μm) and with time step $ d t \u2009 \omega pe = 0.024$ where $ \omega pe$ is the electron plasma frequency. We use 512 and 80 particles per cell per species in 2D and 3D, respectively, for collisionless simulations. In simulations with a binary collision operator, we also use 512 and 80 electron macroparticles per cell in 2D and 3D. The number of ion macroparticles per cell are chosen so that the statistical weight of the individual ion macroparticles is equal to that of the electron macroparticles, ensuring strict energy and momentum conservation in the collision operator.
The multispeckled laser beams have 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.) $ I ave$ is calculated on the injection boundary using the intensity values after the rise time. The laser pulse is launched from the boundary with a polarization in the y direction with the field E_{y} specified in a manner that approximates a Gaussian random field.^{41–44} In vacuum, this creates a random distribution of F/6.7 speckles with characteristic width $ 1.2 F \lambda 0 = 2.8$ μm and length $ 2 \pi F 2 \lambda 0 = 99.0$ μm. The 2D simulation geometry is shown in Fig. 1, illustrating the pump beam, seed beam, and IAW propagation directions. Firstorder Higdon absorbing boundary conditions^{45} are used for the fields.
The CBET behavior is explored in simulations with pump laser beam intensity $ I p = 2.2 \xd7 10 15$ W/cm^{2} (as in Ref. [21]) and either a low seed intensity $ I s = 1.0 \xd7 10 13$ W/cm^{2} or a high seed intensity $ I s = 5.0 \xd7 10 14$ W/cm^{2} with beam crossing angle $ \theta = 99 \xb0$. The beam diameter is 68 μm in 2D and 34 μm in 3D. Except for the reduced beam diameters, the laser and plasma conditions in simulations match those measured in the experiments.^{19,20} Under these laser and initial plasma conditions, the time evolution of CBET is modeled until the gain approaches a steadystate value (except for collisionless simulations at low seed intensity, which were terminated when CBET saturation was clearly seen but before the gain reached a steadystate value). The behavior of CBET gain as a function of the evolving plasma density measured in the experiments is not modeled because the simulations do not include the hydrodynamics of the plasma expansion.
B. SuperGaussian electrons
C. Collisions and pumpbeaminduced thermal effects
Since Coulomb collision rates scale dominantly as $ Z i 2 Z j 2 T \u2212 3 / 2$, where Z_{i} and Z_{j} are the charges of the particles, collisions are important for modeling the OMEGA CBET experiments where the plasma has midZ N^{7+} ions. The collisional simulations are performed using a binary collision model^{50} and can include all self and crossspecies collisions or only selected species collisions. The collision operator uses a fixed Coulomb logarithm $ log \u2009 \Lambda = 7.2$ for all collisions.^{51} To ensure accuracy of collision sampling,^{52} the collision operator is applied every N time steps, where N is defined by the condition $ 2 d t coll \nu ij \u226a 1$, $ d t coll = Ndt$ is the subcycling time step of the collision operator, dt is the simulation time step, and $ \nu ij$ is the collision rate for species i and j (including selfcollisions for i = j and crossspecies collisions for $ i \u2260 j$). The collision model has been validated by comparisons with classical transport theory^{53,54} and has been used successfully in prior studies of LPI experiments.^{19,21,41,55,56}
To understand collisional effects on ion trapping and CBET dynamics, we first examine the collision rates for various scattering processes.^{51} The dominant rates are from N^{7+}, H^{+}, and electron scattering with a background of N^{7+}. Figure 2 shows rates of frictional slowing down ν_{s} (blue curves), perpendicular diffusion $ \nu \u22a5$ (green), and parallel diffusion $ \nu \u2225$ (red) as a function of test particle velocity for N^{7+} (a), H^{+} (b), and electrons (c) propagating in a background of N^{7+} (with temperature 150 eV and number density $ n N = 7.3 \xd7 10 18$ cm^{−3}). The vertical dashed lines indicate the phase velocity of the resonant CBET IAW expressed in units of the test particle thermal speed. This provides a reference for the region of the velocity space distribution where waveparticle interactions occur in VPIC simulations of CBET. Ion trapping modifies the velocity space distribution, IAW damping rate, and CBET gain. Collisional scattering may significantly affect the velocity space distribution in regions around the IAW phase velocity and, as a result, the IAW damping rate and the CBET gain.
For test N^{7+} ions, shown in Fig. 2(a), the slowingdown rate is comparable to the perpendicular diffusion rate, and both rates are much higher than the parallel diffusion rate in the region around the IAW phase velocity. The estimated timescale for slowingdown and perpendicular diffusion is about 10 ps, shorter than a typical VPIC CBET simulation time ( $ \u223c 100$ ps). In CBET, IAW can grow to a large amplitude and trap ions, forming a trapped particle population and modifying the initial Maxwellian distribution function. Collisional scattering processes (mainly slowingdown and perpendicular diffusion) can modify the trapped population and restore the distribution toward a Maxwellian form. The effects of collisions on N^{7+} ions trapped by the IAW can lead to significant N^{7+} ion heating (see Fig. 5 in Ref. 21) as a result of the strong selfcollisions.
For test H^{+} ions, shown in Fig. 2(b), the perpendicular diffusion rate is greater than the slowingdown rate, and both rates are much higher than the parallel diffusion rate in the region around the IAW phase velocity. The estimated timescale for perpendicular diffusion and slowingdown is shorter than 10 ps. The IAW energy is first converted to the directed kinetic energy of the trapped ions. Collisions convert this directed energy into ion thermal energy. However, the partition of IAW energy that gets transferred to each ion species cannot be determined from theory estimates. Nevertheless, we observe from simulations that the H^{+} ions heat less than the N^{7+} ions (also see Fig. 5 in Ref. 21).
As a result of collisions, the ion distributions in the VPIC simulations show a reduced population of trapped ions and increased ion temperatures as time increases. Both the simulated timescale and the amount of ion collisional heating are consistent with experimental observations.^{19–21} Since N^{7+} and H^{+} ion scattering rates on background electrons are much lower, the effects of collisions on ion trapping can be represented in VPIC simulations using only ion–ion collisions as done in Ref. 21.
The CBET IAW may interact with electrons that have a velocity much smaller than the electron thermal speed. For test electrons, shown in Fig. 2(c), the scattering rates increase sharply as the test electron velocity decreases. For the parameters considered here, the main effects that electron collisions have on CBET are fluctuations of both electron temperature and plasma flow.
To understand the thermal effects (i.e., transport effects resulting from electron collisions) on CBET gain, we first consider the dynamics of a highintensity pump beam in the absence of a seed beam by exploring the density response, thermal fluctuations $ \delta T e$, and thermaleffectsinduced flow in the highintensity pump beam with and without collisions. In Fig. 3, we show the plasma density, flow v_{x}, and temperature profiles across the pump beam from simulations at pump intensity $ I p = 2.2 \xd7 10 15$ W/cm^{2}. The laser field $ E y ( x , z )$ is given in frame (a) and frames (b) to (e) show the profiles along x of charge density, flow velocity v_{x}, and temperature from the simulation with full collisions for the electrons (black curves), H^{+} (blue), and N^{7+} (red) ions at an early time and a late time that is long compared with the collision times. Similarly, v_{x} and temperature profiles from the collisionless simulation are shown in frames (f) and (h). The laser light drives perturbations in density, flow velocity, and electron temperature in both the collisional and collisionless simulations. The two dominant transverse gradient scales in these perturbations are the characteristic width of the speckles and the beam width. At early times, the flow perturbations at the speckle scale are comparable in the collisional and collisionless simulations. Later, these smallscale flow perturbations evolve to a larger scale comparable to the beam width. Contrasting the results in frames (c)–(e) with those in (f)–(h), we find enhanced flucturations from thermal effects, as shown by the differences in v_{x} and T_{e} with and without collisions. With full collisions, fluctuations of electron density n_{e} and temperature T_{e} are 10% and 25% of their initial values, respectively. Additionally, thermal effects induce a plasma flow away from the center of the pump beam with a peak v_{x} value ∼5% (for the N^{7+} ions) to 10% (for the H^{+} ions) of the IAW phase velocity $ v ph$. In contrast, in the collisionless case, the fluctuations of electron temperature T_{e} are smaller with peak values at 10% and the plasma flows are also reduced by 2× without thermal effects.
If we were to add a seed beam with a frequency initialized to the resonance condition for CBET in the absence of flow, these modifications to the plasma conditions could affect the CBET matching conditions as well as the CBET IAW damping rates through increased T_{e}. The impact of thermaleffectsinduced flow on the CBET gain can be understood from linear theory: The energy transfer is proportional to the imaginary part of the plasma response function.^{58,59} These flows can Dopplershift the CBET IAW frequency by a few percent, changing the peak of the plasma response function from being on resonance to being off resonance. At the speckle scale, this leads to upshifted frequencies in the left half of the speckles and downshifted frequencies in the right half of the speckles. At the laser beam scale, this leads to upshifted frequencies in the left half of the laser beam and downshifted frequencies in the right half of the laser beam. In the presence of ion trapping, such a Dopplershift away from resonance can be comparable to that caused by trappingmodified distributions, as reported in prior studies of CBET.^{21,29} In Sec. III below, we examine the combined detuning effects from ion trapping and flow on the nonlinear saturation of CBET.
III. NONLINEAR SATURATION OF CBET BY ION TRAPPING AND THERMALEFFECTSINDUCED FLOW
CBET dynamics can be influenced by ion trapping and ionheatinginduced detuning,^{21} as well as by flow induced by thermal effects as explained above. Collisions may affect both processes: Collisional detrapping is important near the onset threshold intensity for trapping; collisions also introduce thermal effects, enhancing the plasma flow. Since ionelectron collision rates are small compared with Ion–ion collision rates (see Sec. II C), only ion–ion collisions were included in prior CBET simulations^{21} to isolate the effects of ion trapping and heating from those of evolving plasma conditions. Here VPIC simulations were run with full collisions, ion–ion collisions, and no collisions to determine the effect of collisions on iontrapping and flow modifications in CBET.
In this work, the CBET gain percentage of the seed beam is defined 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 input (measured on the right boundary) and output power (measured on the left boundary) of the seed beam, respectively. The gain percentage can be converted to gain $ G = ln \u2009 ( P s \u200a out / P s \u200a in )$ as used in linear theory.
FSRS may be unstable in the CBET amplified seed beam, as found in VPIC simulations relevant to indirectdrive.^{28,29} However, FSRS is found to be unimportant in this work because of the low plasma densities used in the OMEGA CBET experiments. In the absence of FSRS, ion trapping is the main nonlinear effect aside from detuning via flow fluctuations and pump depletion.
We start with a discussion of results from 2D simulations, followed by 3D simulations that confirm the main results.
A. 2D VPIC simulations
In Fig. 4, the CBET gain vs time from collisionless, ion–ion collision, and fully collisional 2D simulations are shown at two seed intensities: low seed intensity $ I s = 1.0 \xd7 10 13$ W/cm^{2} (red curves) and high seed intensity $ I s = 5.0 \xd7 10 14$ W/cm^{2} (black, blue, and green curves). Simulations using different speckle patterns show that the CBET gains are insensitive to speckle statistics in the cases examined. At low seed beam intensity, the CBET gain is sensitive to collisions as evidenced in the substantially different CBET gains shown by the three red curves. In contrast, at high seed intensity, the CBET gain is insensitive to collisions shown by the black, blue, and green curves.
Differences in the CBET gain can arise from the following: (1) Full collisions lead to changes in n_{e} and T_{e} and modify the plasma conditions for CBET resonance. An increase in T_{e} can also lead to a reduction of IAW damping. (2) Full collisions enhance laserinduced flow, as shown in Fig. 3, which also modifies the CBET resonance. (3) Collisional detrapping has a significant effect on the CBET gain when ion trapping is relatively weak.
First, we use the low seed intensity cases to explain the effects of collisional detrapping of ions. In Fig. 5, at low seed intensity $ I s = 1.0 \xd7 10 13$ W/cm^{2}, the characteristic ion 2D velocity space distributions $ f ( v x , v z )$ are compared from collisionless, ion–ion collision, and fully collisional simulations shown in frames (a1) and (b1), (a2) and (b2), and (a3) and (b3), respectively. These distributions are sampled at a time (t = 90 ps) during CBET saturation but are generally representative of the entire durations of the simulations. The trapping tail, which indicates the IAW propagation direction (see Fig. 1), is the strongest for the collisionless case in Figs. 5(a1) and 5(b1). In simulations with collisions, collisional detrapping and ion heating resulting from collisional scattering can effectively reduce the trapping tail in the velocity space distributions. As explained in Sec. II C, the N^{7+} and H^{+} ion scattering rates on background electrons are much lower than ion–ion scattering rates. Thus, the effects of collisions on ion trapping are similar in simulations with ion–ion collisions and with full collisions, as seen by comparing Figs. 5(a2) and 5(b2) to Figs. 5(a3) and 5(b3). The CBET gain reduction from the solid red curve to the dashed red curve in Fig. 4 results from ion–ion collisional detrapping alone since the ponderomotiveeffectsinduced flow is the same in collisionless and ion–ion collision simulations. We note that while electron trapping in the IAW and its induced nonlinearity (i.e., a positive nonlinear frequency shift as opposed to the negative frequency shift associated with ion trapping) have been reported with large $ Z T e / T i > 10$,^{60,61} they do not present discernible effects in our CBET simulations.
In Fig. 6, at low seed intensity $ I s = 1.0 \xd7 10 13$ W/cm^{2}, 2D simulation results for flow velocity v_{x} and temperature are compared for collisionless [frames (a1)–(d1)], ion–ion collision [frames (a2)–(d2)], and fully collisional [frames (a3)–(d3)] cases. As we explained in Sec. II C, laserdriven flow at the speckle scale can begin to detune the CBET resonance at early times of the interaction. Here, we focus on flow induced by thermal effects at long time scales. Profiles of v_{x} and temperature along x (at z = 0.0 μm) for electrons (black curves), H^{+} (blue) and N^{7+} (red) ions are shown at t = 50.0 ps (near the peak of CBET gain) and at 91.5 ps (during saturation). Note that in the presence of ion trapping, taking the first moment of the distribution function yields an apparent flow with all positive values in velocity $ v x ( x )$ even though there is no initial flow in the simulation. Furthermore, taking the second moment of the distribution function yields an effective temperature, which is higher than the initial ion temperature. Both arise from trapping as seen in Fig. 5. In the collisionless and ion–ion collision simulations, the ponderomotiveeffectsinduced flow is small and the ion velocity $ v x ( x )$ is predominantly from the modification of the velocity space distribution due to iontrapping. The effects of detrapping can be discerned from $ v x ( x )$ and the ion temperature profiles by comparing the collisionless and the ion–ion collision simulations: with ion–ion collisional detrapping, the amplitudes of v_{x} and ion temperatures are lower in the ion–ion collision case.
Next, we examine how flow induced by thermal effects accounts for the reduction in the CBET gain from the dashed red curve to the dotted red curve in Fig. 4. As shown in Fig. 3, thermal effects in the intense pump beam induce a plasma flow outward from the axis of the pump beam with a peak value ∼5% (for the N^{7+} ions) to 10% (for the H^{+} ions) of the IAW phase velocity $ v ph$. In the simulation with full collisions, as a result of the combined effects from ion trapping, collisional ion heating, and thermaleffectsinduced flow in the pump beam, the profile of $ v x ( x )$ is asymmetrical about the center of xdomain (x = 69 μm), i.e., the magnitude of the flow is larger to the right of x = 69 μm than to the left. This asymmetry decreases in time as trapped ions in the tails of the distributions go through slowingdown, perpendicular diffusion, and thermalizing processes, leading to a flow $ v x ( x )$ resembling that in the pumponly simulation in Fig. 3. This induced flow further detunes CBET away from resonance in addition to the ion trapping/heating induced detuning, leading to the CBET gain reduction from the dashed red curve to the dotted red curve in Fig. 4. We note that in the simulation with full collisions, there is also heating in T_{e} up to 15%.
At the high seed intensity $ I s = 5.0 \xd7 10 14$ W/cm^{2}, in Fig. 7, 2D simulation results for flow velocity v_{x} and temperature are compared for ion–ion collision [frames (a1) and (b1)] and full collision [frames (a2) and (b2)] cases. The ion trapping is strong and leads to nearly all positive values of $ v x ( x )$ and large ion temperature increases. The ion trapping induced detuning dominates over effects from collisional detrapping and flow. These results explain why the CBET gain at high seed intensity is insensitive to collisional detrapping and thermaleffectsinduced flow, as shown by the lower black solid, blue dashed, and green dotted curves in Fig. 4. Ion trapping induced saturation leads to a reduction of the CBET gain to a level that is much lower than seen at low seed intensity, as shown in Fig. 4, or as predicted by linear theory including pump depletion.^{19,21}
Note that for the 2D simulations, the average intensity for the speckled pump and seed beams is defined $ I ave \u2261 \u27e8 I 2 \u27e9 / \u27e8 I \u27e9$, as used in Omega experiments,^{62} where brackets denote spatial averages over the beam cross section at the laser launch surface. Converting the intensity to that based on the normal average intensity definition $ \u27e8 I \u27e9$ and using simple linear theory estimates,^{21} we find that the gain percentage from linear theory is $ \u223c 49$% (using the interaction length $ L = d / \u2009 sin \u2009 \theta $ where d is the beam diameter and θ is the beam crossing angle). This value is below those from simulations at low seed intensity but above those from simulations at high seed intensity.
B. 3D VPIC simulations
To determine the effects of dimensionality and to confirm the main findings from 2D simulations, additional 3D simulations are performed with and without collisions at low and high seed intensities. The simulation domain is 70.4 $ \xd7 17.3$ $ \xd7 54.0$ μm^{3} in x, y, z with periodic boundary conditions in y. The pump and probe laser beams on the x and z launch boundaries have rectangular cross sections of dimension $ 34 \xd7 17.3$ μm^{2} in their respective launch planes. With more than 5 × 10^{9} cells and 80 particles per cell, the simulations employ 211 600 MPI ranks and over 1.2 × 10^{12} particles, which reaches the limit of what is practical with the computing resources available to us. However, at reduced scale, for a random distribution of F/6.7 speckles with a characteristic width $ 1.2 F \lambda 0 = 2.8$ μm and length $ 2 \pi F 2 \lambda 0 = 99.0$ μm, these simulations represent the multispeckled nature of the 3D CBET interactions. The 3D geometry allows for additional laserdriven flow and temperature fluctuations in y as well as modifications to ion trapping such as enhanced side loss compared with the 2D simulations presented in Sec. III A.
The time histories of the CBET gain from the 3D simulations at low seed intensity $ I s = 1.0 \xd7 10 13$ W/cm^{2} and high seed intensity $ I s = 5.0 \xd7 10 14$ W/cm^{2} are shown in Fig. 8(a). (Note that here, in contrast with 2D simulations, we use the more traditional definition of intensity $ I s \u2261 \u27e8 I \u27e9$, where brackets denote a spatial average over the beam cross section at the launch plane.) The red solid curve and the red dotted curve are at low seed intensity without collisions and with full collisions, respectively. Similar to the 2D simulation results shown in Fig. 4, the 3D simulations also show that at low seed beam intensity, the CBET gain is sensitive to collisions. This sensitivity, as explained using results from the 2D simulations, arises from collisional detrapping and thermaleffectsinduced flow, as well as from changes in the plasma conditions in the presence of full collisions.
To compare with the thermal effects in 2D, results from the 3D simulation with full collisions at low seed intensity are shown at a time t = 64.2 ps during CBET saturation in Figs. 8(b)–8(d). Profiles along x (averaged over 10 cells around $ y = z = 0$) for charge density (b), flow velocity v_{x} (normalized to the IAW phase velocity $ v ph$) (c), and temperature (normalized to the initial temperature for each species) (d) are shown for the electrons (black curves), H^{+} (blue) and N^{7+} (red) ions. Thermal effects in the intense pump beam are stronger in 3D, as evidenced by the stronger heating of T_{e} in 3D with an increase in T_{e} up to 30%, in contrast with that in Fig. 6(d3). The asymmetry in the profile of $ v x ( x )$ about the center of the xdomain (x = 35 μm) reflects the presence of induced flow superimposed onto an allpositive valued $ v x ( x )$ profile due to ion trapping alone. The net profile $ v x ( x )$ has a larger magnitude in the x > 35 μm region than in the x < 35 μm region.
For high seed intensity, the time histories of CBET gain from 3D simulations are shown in Fig. 8(a) by the black solid and green dotted curves without collisions and with full collisions, respectively. Profiles along x for flow velocity v_{x} (e) and temperature (f) are also shown for the different species. The strong ion trapping effects dominate over those from thermal effects, and the values of $ v x ( x )$ are largely positive. In contrast to the low seed intensity case, at high seed intensity, the CBET gain is insensitive to collisions as well as dimensionality as a result of the dominant ion trapping effects.
Though 2D and 3D simulations of CBET exhibit similar behavior in many regards, there are also important differences. In 3D, the most intense laser speckles in the crossing beams, for which CBET IAW amplitudes are largest, are less likely to overlap than in 2D, where essentially every intense speckle in the pump and seed beams overlap. One would also expect weaker transverse coupling among neighboring speckles through the exchange of sideloss ions as compared with 2D. The sideloss ions originate from the trapped ions that escape the narrow IAW potential well^{63} at a rate $ \u223c v th , i / F \lambda 0$. The sideloss ions can lower the Landau damping rate for IAW as they enter neighboring speckles, thereby increasing the wave coherence length and enhancing energy transfer.^{28} In 2D, every sideloss ion leaving a given speckle will encounter a neighboring speckle (except for speckles at the edge of a beam). In 3D, only a fraction of the sideloss ions leaving a given speckle encounter a neighboring speckle, which reduces the transverse coupling among neighboring speckles.
Another interesting result of our study is the apparent effects of boundary conditions on the CBET gain at low seed intensity. Comparing the red solid and dotted curves in Fig. 4 (with larger beam diameter) to their counterparts in Fig. 8(a) (with smaller beam diameter), we see that although the peak of the gain curve from the 3D collisionless simulation is lower than that from its 2D counterpart, the 3D gain saturates at a higher level than that of the 2D simulation; for fully collisional simulations, the gain from 3D is also higher than that from 2D. This is because in the 2D simulations, all physical boundaries are open to waves and particles. As a result, energy from the laser beams enters and exits the boundaries and particle and wave energy from the CBET processes also exit the boundaries, which act as an energy sink. In particular, ions accelerated by waveparticle interactions and electrons heated by the laser can propagate out of the boundaries and are replenished by Maxwellian and superGaussian ions and electrons at initial temperatures flowing into the system from the boundaries. The 3D simulations allow for physical processes, such as ion trapping and laserdriven flow and temperature fluctuations in the y dimension, but the boundary conditions are periodic in y. Ions accelerated by waveparticle interactions and electrons heated by the laser are wrapped around in the y domain, which is effectively infinite. It is unclear in this case what the correct interaction length is in estimating the gain percentage using simple linear theory.^{21}
IV. SUMMARY AND CONCLUSIONS
Using 2D and 3D VPIC simulations for parameters relevant to recent CBET experiments at the OMEGA laser facility, the influences on CBET of ion trapping in ion acoustic waves and laserdriven flow and temperature fluctuations have been examined. In the midZ plasma with an intense pump laser beam, ion trapping, collisional detrapping, and plasma flow induced by thermal effects all have important effects on CBET. Ion trapping can enhance or detune the CBET resonance. At low seed beam intensity near the onset threshold for ion trapping, collisional detrapping has more significant effects on CBET and thermal effects can induce flows, which also detune the CBET resonance at a level comparable to that from trapping. At high seed beam intensity, ion trapping effects on CBET dominate over those from collisional detrapping and the flow induced by thermal effects. The trapping dominated CBET saturation has been confirmed from simulations at seed intensities ranging from 3 to $ 5 \xd7 10 14$ W/cm^{2}. As a consequence, we found that CBET gain is sensitive to collisions and dimensionality at low seed beam intensity where ion trapping is weak but is insensitive to collisions and dimensionality at high seed beam intensity with strong ion trapping. In both the high and lowintensity seed cases, the amount of energy gained by the seed beam represents a very small fraction of the pump and the effects of pump depletion are minimal.
We note that the pump intensity used in this study ( $ 2.2 \xd7 10 15$ W/cm^{2}) is higher than the intensity of the individual pump beams of the experiments, though matches the pump intensity used in our prior CBET modeling study.^{21} This choice allows us to compare the simulation results and isolate the thermal effects in the presence of ion trapping. These effects include the introduction of a spatially varying (over lengths comparable to the width of a laser speckle or the beam) transverse flow whose speed is a small fraction of the IAW speed. Such a flow may be difficult to measure experimentally, yet may nevertheless affect CBET gain. An additional pumponly simulation with full collisions at a lower pump intensity $ I p = 1.0 \xd7 10 15$ W/cm^{2} has also been performed and the plasma flow v_{x} profiles are shown in Fig. 3(d) by the orange and green curves for N^{7+} and H^{+} ions, respectively, with the peak v_{x} values ∼4% to 5% of the IAW phase velocity $ v ph$ (compared to the ∼5% to 10% at $ I p = 2.2 \xd7 10 15$ W/cm^{2} that can Dopplershift the CBET IAW frequency by a few percent and detune the peak of the plasma response function from being on resonance to being off resonance). The results suggest that the intensity threshold for thermal effects to affect CBET gain is near $ I p = 1.0 \xd7 10 15$ W/cm^{2}. For these simulations, the average intensity for the speckled beam defined as $ I ave \u2261 \u27e8 I 2 \u27e9 / \u27e8 I \u27e9$ are used. Converting the intensity to that based on the normal average intensity definition $ \u27e8 I \u27e9$, the pump beam intensities for these simulations are $ I p = 1.32 \xd7 10 15$ and $ 6 \xd7 10 14$ W/cm^{2}.
The electron temperature increases at 25%–30% resulting from laser heating shown in Figs. 3 and 8 occur over a time interval that is long compared to collision times but short compared to the 50 ps temporal resolution of the Thomsonscattering system measurements.^{19} Together with the $ P \u2009 d V$ cooling associated with the slowly decreasing density observed in the experiments, the electron heating may be difficult to observe experimentally.
One of the main results of this work is the elucidation of important effects of ion trapping on CBET saturation. This has broad applicability to fusionrelevant laserplasma configurations as discussed in prior work.^{26,27} Collisional detrapping and plasma flow induced by thermal effects also have wide applicability to ICF and HED settings. The choice of parameter regime for the simulations (and the Top9 experiments) has relevance to other ICF and HED settings as well. The plasma composition and $ Z *$ used in this work resembles that of CH target in directdrive ICF implosion. Also, in the low plasma density regime, as in the case for low gas fill and nearvacuum hohlraums,^{66–70} FSRS and BSBS are not dominant instabilities, so ion trapping is the main nonlinear effect aside from detuning via flow fluctuations and pump depletion, and its effects can be examined in isolation. Key parameters determining the CBET gain include $ Z * T e / T i$, IAW damping rate, and IAW $ k \lambda D$ (or IAW phase velocity normalized to the ion thermal velocity).^{29} Although in the regime of low density, the simulations in this work (and the Top9 experiments) can access different regimes of IAW $ k \lambda D$ and IAW damping rates by varying the crossing angle and density. The resulting values of the IAW damping rates are relevant to those in OMEGA and NIF scale directdrive experiments.^{21} Moreover, this work reports on key processes in the nonlinear optics of plasmas, identified as a grand challenge problem in plasma and high energy density physics. As such, these results would be of broad interest to the plasma physics community. For example, results from recent beam combiner experiments indicate that the energy output in the seed beam may be increased further by a reduction in the plasma density^{64,65} if the nonlinear saturation of CBET were mitigated.
We further note that the kinetic simulations for our study were performed under initial conditions of no bulk plasma flow but with a frequency difference between the pump and seed beams giving rise to the excitation of CBET. In the presence of plasma flow, CBET can also occur between beams of the same frequency in regions of sonic flow.^{38,39,71} Under these conditions, thermaleffectsinduced flow would be expected to have similar detuning effects on CBET, a point we intend to consider in future work.
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 and was supported by the LANL Office of Experimental Sciences Inertial Confinement Fusion program. LLE work was supported by the Department of Energy National Nuclear Security Administration under Award No. DENA0003856. The authors acknowledge valuable discussions with Dr. Harvey Rose. VPIC simulations were run on ASC Trinity supercomputer under ATCC and the Large Scale Calculations Initiative (LSCI).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Lin Yin: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). K. L. Nguyen: Investigation (supporting); Writing – review & editing (supporting). Brian J. Albright: Conceptualization (supporting); Investigation (supporting); Methodology (equal); Software (equal); Writing – review & editing (supporting). Alexander G. Seaton: Investigation (supporting); Methodology (supporting); Software (equal); Writing – review & editing (supporting). Aaron Hansen: Investigation (supporting); Writing – review & editing (supporting). Dustin Froula: Investigation (supporting); Writing – review & editing (supporting). David Turnbull: Investigation (supporting); Writing – review & editing (supporting). John Patrick Palastro: Investigation (supporting); Writing – review & editing (supporting).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.
APPENDIX: SUPERGAUSSIAN ELECTRONS

Generate four samples p_{1}, p_{2}, p_{3}, and p_{4} independently from a uniform random distribution from zero to one.

Pick a sample speed $ u samp = \u2212 log \u2009 p 1$, obtained from inverting $ f env ( u )$.

If $ p 2 f env ( u samp ) \u2264 u samp 2 \u2009 f 3 ( u samp )$, accept the sample and proceed to step 4. Otherwise, reject the sample and return to step 1.
 Generate the components of u by sampling uniformly over solid angle$ \mu = 2 ( p 3 \u2212 1 2 ) \varphi = 2 \pi p 4 u x = u samp \u2009 cos \u2009 \varphi 1 \u2212 \mu 2 u y = u samp \u2009 sin \u2009 \varphi 1 \u2212 \mu 2 u z = u samp \mu .$

Generate two samples p_{1} and p_{2} independently from a uniform random distribution from zero to one.

Pick a sample u value $ u samp = \u2212 log \u2009 p 1$, obtained from inverting $ f env ( u z )$.

If $ 2.83 p 2 f env ( u samp ) > u samp f 3 , z fit ( u samp )$, reject the sample and return to 1.
This prescription rejects approximately one third of samples and has proven to be acceptably efficient.

Generate three samples p_{1}, p_{2}, and p_{3} independently from a uniform random distribution from zero to one.

Pick a sample u value $ u samp = \u2212 log \u2009 p 1$, obtained from inverting $ f env ( u z )$.

If $ 1.165 \u2009 exp \u2009 ( \u2212 u z 3 ) p 2 f env ( u samp ) > u samp \u2009 exp \u2009 [ \u2212 ( u z 2 + u samp 2 ) 3 / 2 ]$, reject the sample and return to 1.
 Set $ \varphi = 2 \pi p 3$ and set the two tangential components $ u t 1 , \u2009 u t 2$ to$ u t 1 = u samp \u2009 cos \u2009 \varphi u t 2 = u samp \u2009 sin \u2009 \varphi .$
This sampling rejects a fraction 0.41 of trials on average. For numerical stability, it is recommended that u_{z} be rejected above u_{z} = 4 prior to entering this portion of the sampling algorithm. Overall, for VPIC calculations on a variety of manycore supercomputers available to us, the Maxwellian reinjection of superGaussian distributions adds of order 10%–20% overhead to the runtime, making these runs practicable.