The influences of ion trapping and fluctuations of electron temperature and plasma flow on cross-beam energy transfer (CBET) are examined using two- and three-dimensional particle-in-cell simulations in parameter regimes relevant to recent CBET experiments at the OMEGA laser facility. In mid-Z plasma irradiated by an intense pump beam and weaker probe beam, ion trapping, collisional de-trapping, 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 de-trapping can affect the CBET gain at low seed beam intensity near the onset threshold for ion trapping. Thermal-effects-induced 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.

Laser-plasma instabilities (LPI) reduce the energy coupling to plasma and can affect both the direct- and indirect-drive approaches to laser-driven 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 electron-plasma wave (EPW) and an ion-acoustic wave (IAW), respectively.7 The cross-beam 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 multi-physics 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 facility19,20 at the Laboratory for Laser Energetics (LLE) were designed to provide a simplified, well-diagnosed 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 self-consistent, nonlinear, particle-in-cell (PIC) modeling. A recent study of this type led to an improved understanding of nonlinear CBET phenomena.21 Using two-dimensional (2D) VPIC22–25 simulations, Nguyen et al. found that CBET can saturate through ion trapping-induced 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 off-resonant 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 laser-beam self-focusing. One form of self-focusing is ponderomotive self-focusing, arising when plasma density depressions form in high-intensity beams as a result of plasma hydrodynamics response to the ponderomotive pressure.7 The other is thermal self-focusing from plasma heating by the inverse bremsstrahlung.30–37 CBET in the presence of speckled-laser beam self-focusing has been examined in studies using a fluid model of the plasma.38,39 It was found that ponderomotive self-focusing in speckled beams has a significant impact on CBET resulting from beam bending and plasma-induced smoothing. However, the hydrodynamic equations employed an isothermal approximation and, thus, thermal effects37 in laser self-focusing 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 large-scale speckled beams28,29 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 three-dimensional (3D) VPIC simulations of a seed beam interacting with a pump beam in parameter settings relevant to the OMEGA CBET experiments, namely, in a mid-Z plasma with an intense pump laser beam. To the best of our knowledge, these are the first ever multi-speckled 3D fully kinetic simulations of CBET. We focus here on the influences of ion trapping and laser-induced thermal effects. In Sec. II, we describe the PIC simulation setup for modeling the OMEGA CBET experiments, including the methods for modeling the super-Gaussian electron distribution, inter-particle collisions, and thermal effects. In Sec. III, the effects of ion trapping, collisional de-trapping, 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 full-collisions simulations in 2D and 3D. The main results from this work are summarized in Sec. IV.

The plasma and laser conditions for the VPIC simulations were chosen to correspond to those of recent CBET experiments19,20 using the Tunable OMEGA Port-9 laser.40 The platform employs a gas-jet 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 3.2 × 10 15 W/cm2) and probe beam (with intensities ranging from 0.1 to 4.1 × 10 14 W/cm2) 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 ° over an interaction length of 220 μm. The pump and probe beams were linearly co-polarized. Using Thomson-scattering measurements, the plasma was observed to be spatially uniform over a 1.5 mm diameter plateau at initial temperatures Te = 450 eV, T i = T e / 4, and density n e = 8 × 10 19 cm−3, though the density was observed to decrease slowly to 5.5 × 10 19 cm−3 over a period of 700 ps.

In the VPIC simulations, the plasma has electrons and pre-ionized hydrogen and nitrogen ion species H+ and N7+ (with realistic mass ratios) at 55% and 45% number density, respectively, with initial Maxwellian ion and non-Maxwellian electron (super-Gaussian of order three) distributions, matching the experiments.19,20 We model the experiments with a spatially uniform, representative density n e = 6 × 10 19 cm−3 with n e / n cr = 0.006 (where n cr is the critical density) and electron-to-ion temperature ratio T e / T i = 4 and Te = 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 super-Gaussian of order three for the electrons at initial temperatures Ti and Te.

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 λD (0.02 μm) and with time step d t ω pe = 0.024 where ω 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 macro-particles per cell in 2D and 3D. The number of ion macro-particles per cell are chosen so that the statistical weight of the individual ion macro-particles is equal to that of the electron macro-particles, ensuring strict energy and momentum conservation in the collision operator.

The multi-speckled laser beams have an average intensity I ave and a flat-top 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 Ey 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 λ 0 = 2.8 μm and length 2 π F 2 λ 0 = 99.0 μm. The 2D simulation geometry is shown in Fig. 1, illustrating the pump beam, seed beam, and IAW propagation directions. First-order Higdon absorbing boundary conditions45 are used for the fields.

FIG. 1.

2D simulation geometry showing the pump and seed beam laser intensity | E y ( x , z ) | 2 launched from the top and the right boundaries, respectively. The beam crossing angle θ = 99 ° and the beam diameters are 68 μm. The IAW propagation direction is as indicated.

FIG. 1.

2D simulation geometry showing the pump and seed beam laser intensity | E y ( x , z ) | 2 launched from the top and the right boundaries, respectively. The beam crossing angle θ = 99 ° and the beam diameters are 68 μm. The IAW propagation direction is as indicated.

Close modal

The CBET behavior is explored in simulations with pump laser beam intensity I p = 2.2 × 10 15 W/cm2 (as in Ref. [21]) and either a low seed intensity I s = 1.0 × 10 13 W/cm2 or a high seed intensity I s = 5.0 × 10 14 W/cm2 with beam crossing angle θ = 99 °. 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 steady-state value (except for collisionless simulations at low seed intensity, which were terminated when CBET saturation was clearly seen but before the gain reached a steady-state 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.

To model the OMEGA CBET experiments,46,47 we initialize the electron velocities using non-Maxwellian (super-Gaussian) electron velocity distribution functions with particle-refluxing boundary conditions that reinject these distributions at the boundary. The distribution functions are of the form reported by Matte et al.48 
f m ( v ) d 3 v = C m exp [ ( v v m ) m ] d 3 v ,
(1)
where v | v |,
v m = ( k B T e m e ) 1 / 2 [ 3 Γ ( 3 / m ) Γ ( 5 / m ) ] 1 / 2 ,
(2)
C m = 1 4 π ( 2 v th , e 2 ) 3 / 2 m a m 3 / 2 Γ ( 3 / m ) ,
(3)
and the Gamma function is defined as
Γ ( n ) 0 d t t n 1 e t .
(4)
In the  Appendix, we describe an efficient rejection sampling method for the particle load and reinjection for the special case m = 3, as measured in the OMEGA CBET experiments.
The pump beam has wavelength λ 0 = 351 nm. To determine the seed beam frequency, we use the electron susceptibility for an m = 3 super-Gaussian distribution in a root-finding solver to obtain the IAW frequency. This is found from the real part of the frequency ω from solving the ion acoustic wave dispersion relation ε ( ω , k ) = 1 + χ e ( ω , k ) + χ i ( ω , k ) = 0 for IAW wavenumber k = | k p k s |. The seed frequency is given by ω s = ω p ω, where subscripts “s” and “p” stand for seed and pump, respectively, and ω and k are for the IAW. The IAW k is determined from the beam crossing angle and plasma conditions and k = 2 k 0 sin ( θ / 2 ), where k0 is the pump laser wavenumber. The light wave k 0 λ D is determined from the dispersion relation k 0 λ D = ( v th , e / c ) ( ω 0 / ω pe ) 2 1, where v th , e = T e / m e is the electron thermal speed, ω pe = ( 4 π n e e 2 ) / m e is the electron plasma frequency, and Te, ne, e, and me are electron temperature, density, charge, and mass, respectively. The IAW dispersion relation includes slow and fast modes for the two ion-species plasma. In the case modeled here, k λ D = 0.64, the slow mode has the lowest damping rate, and the frequency on the slow mode dispersion branch is used to determine the seed beam frequency. As shown in the  Appendix, the electron susceptibility for m = 3 takes the form
χ e ( ω , k ) = [ Γ ( 5 3 ) ] 3 / 2 2 3 v th , e 3 ω pe 2 k 2 d v v exp [ ( v 2 / v 3 2 ) 3 / 2 ] v ( ω / k )
(5)
and the ion susceptibility is expressed in the usual way in terms of the plasma dispersion function.49 

Since Coulomb collision rates scale dominantly as Z i 2 Z j 2 T 3 / 2, where Zi and Zj are the charges of the particles, collisions are important for modeling the OMEGA CBET experiments where the plasma has mid-Z N7+ ions. The collisional simulations are performed using a binary collision model50 and can include all self- and cross-species collisions or only selected species collisions. The collision operator uses a fixed Coulomb logarithm log Λ = 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 ν ij 1, d t coll = Ndt is the sub-cycling time step of the collision operator, dt is the simulation time step, and ν ij is the collision rate for species i and j (including self-collisions for i = j and cross-species collisions for i j). The collision model has been validated by comparisons with classical transport theory53,54 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 N7+, H+, and electron scattering with a background of N7+. Figure 2 shows rates of frictional slowing down νs (blue curves), perpendicular diffusion ν (green), and parallel diffusion ν (red) as a function of test particle velocity for N7+ (a), H+ (b), and electrons (c) propagating in a background of N7+ (with temperature 150 eV and number density n N = 7.3 × 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 wave-particle 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.

FIG. 2.

Rates of frictional slowing down νs (blue), perpendicular diffusion ν (green), and parallel diffusion ν (red) as a function of test particle velocity (normalized to the thermal speed) for N7+ (a) and H+ test ions (b), and test electrons (c) propagating in a background of N7+ with temperature 150 eV and number density n N = 7.3 × 10 18 cm−3. The v th is defined as the test particle thermal velocity if it were at the same temperature as the background ( v th = T / m). The vertical dashed lines indicate the phase velocity of the IAW driven in the CBET geometry considered here.

FIG. 2.

Rates of frictional slowing down νs (blue), perpendicular diffusion ν (green), and parallel diffusion ν (red) as a function of test particle velocity (normalized to the thermal speed) for N7+ (a) and H+ test ions (b), and test electrons (c) propagating in a background of N7+ with temperature 150 eV and number density n N = 7.3 × 10 18 cm−3. The v th is defined as the test particle thermal velocity if it were at the same temperature as the background ( v th = T / m). The vertical dashed lines indicate the phase velocity of the IAW driven in the CBET geometry considered here.

Close modal

For test N7+ ions, shown in Fig. 2(a), the slowing-down 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 slowing-down and perpendicular diffusion is about 10 ps, shorter than a typical VPIC CBET simulation time ( 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 slowing-down and perpendicular diffusion) can modify the trapped population and restore the distribution toward a Maxwellian form. The effects of collisions on N7+ ions trapped by the IAW can lead to significant N7+ ion heating (see Fig. 5 in Ref. 21) as a result of the strong self-collisions.

For test H+ ions, shown in Fig. 2(b), the perpendicular diffusion rate is greater than the slowing-down 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 slowing-down 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 N7+ 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 N7+ 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.

Lushnikov and Rose analyzed thermal self-focusing using a paraxial equation
( i z + 1 2 k 0 2 k 0 n e 2 n c ρ ) E = 0 ,
(6)
to evolve the electric field envelope E. In their notation, = x ̂ x + y ̂ y denotes a gradient perpendicular to the light wave propagation direction in z ̂, ne is the background electron density, ρ = δ n e / n e is the density fluctuation, k 0 = 2 π / λ 0 is the laser wavenumber in vacuum, and nc is the critical density. They coupled this equation to a linearized hydrodynamic description of the density fluctuation
( 2 t 2 + 2 ν ̃ t c s 2 2 ) ln ( 1 + ρ ) = c s 2 2 ( I + δ T e T e ) ,
(7)
where cs is the ion acoustic wave speed and ν ̃ is an integral operator with perpendicular Fourier transform ν i a k c s, where νia is the ion acoustic wave damping rate normalized to ion acoustic frequency. From this expression, we see that thermal fluctuations seed self-focusing in a manner similar to the ponderomotive term. The density fluctuations result in a change in the index of refraction, which modifies the beam propagation. In Ref. 37, thermal self-focusing was shown to result in beam spray over large beam propagation distances. In our study, we are primarily concerned with a more subtle effect, namely, how thermal fluctuations seed plasma flow in a direction transverse to the laser beam and speckles, which in turn may Doppler-shift and de-tune the IAW resonance. These flows are expected to have speeds that are a small fraction of the IAW speed, varying in space over distances of order the transverse correlation length of the laser field or the beam width and, thus, could be challenging to observe directly in experiments.
Lushnikov and Rose also showed through analysis of a reduced Epperlein model33 that thermal corrections become important when the inverse bremsstrahlung heating parameter g satisfies
g ( k λ e ) = [ 1 + ( 30 k λ e ) 4 / 3 ] 96 ( k λ e ) 2 Z * 1 ,
(8)
where k is the wavenumber of the electron temperature fluctuation, λ e = ( λ e i / 3 ) ( 2 Z * / π ϕ ) 1 / 2, λei is the electron-ion collisional mean free path (which is comparable to the speckle width in our study), Z * = i n i Z i 2 / i n i Z i is the average ion charge, and ϕ = ( 4.2 + Z * ) / ( 0.24 + Z * ) is an empirical factor.57 For our study, as evidenced in Fig. 3, the transverse gradients in electron temperature are dominated by the k associated with the characteristic width of the speckles k k 0 / F. As g ( k λ e ) = 2.3, thermal self-focusing is expected to be comparable to or larger than ponderomotive self-focusing. Another dominant transverse wavenumber for temperature variation is the inverse of the beam width, 1 / w. Since g ( 2 π λ e / w ) 25, thermal self-focusing would dominate over ponderomotive self-focusing over these scales as well.
FIG. 3.

Pump-only 2D simulation at intensity I p = 2.2 × 10 15 W/cm2 with full collisions but in the absence of CBET showing the pump beam laser field E y ( x , z ) (a), profiles along x (averaged over 20 cells around z = 0 at time t = 64.6 ps) of charge density (b), flow velocity vx (first moment of the distribution functions normalized to the IAW phase velocity v ph) (c) and (d), and temperature (second moment of the distribution functions in the center of mass frame normalized to the initial temperature for each species) (e) for electrons (black curves), H+ (blue), and N7+ (red) ions. In (d), the green and orange curves are for H+ and N7+ ions, respectively, at a lower pump intensity I p = 1.0 × 10 15 W/cm2 for discussion in Sec. IV. As a reference, frames (f)–(h) show the profiles of vx and temperature from a pump-only simulation I p = 2.2 × 10 15 W/cm2 without collisions.

FIG. 3.

Pump-only 2D simulation at intensity I p = 2.2 × 10 15 W/cm2 with full collisions but in the absence of CBET showing the pump beam laser field E y ( x , z ) (a), profiles along x (averaged over 20 cells around z = 0 at time t = 64.6 ps) of charge density (b), flow velocity vx (first moment of the distribution functions normalized to the IAW phase velocity v ph) (c) and (d), and temperature (second moment of the distribution functions in the center of mass frame normalized to the initial temperature for each species) (e) for electrons (black curves), H+ (blue), and N7+ (red) ions. In (d), the green and orange curves are for H+ and N7+ ions, respectively, at a lower pump intensity I p = 1.0 × 10 15 W/cm2 for discussion in Sec. IV. As a reference, frames (f)–(h) show the profiles of vx and temperature from a pump-only simulation I p = 2.2 × 10 15 W/cm2 without collisions.

Close modal

To understand the thermal effects (i.e., transport effects resulting from electron collisions) on CBET gain, we first consider the dynamics of a high-intensity pump beam in the absence of a seed beam by exploring the density response, thermal fluctuations δ T e, and thermal-effects-induced flow in the high-intensity pump beam with and without collisions. In Fig. 3, we show the plasma density, flow vx, and temperature profiles across the pump beam from simulations at pump intensity I p = 2.2 × 10 15 W/cm2. 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 vx, and temperature from the simulation with full collisions for the electrons (black curves), H+ (blue), and N7+ (red) ions at an early time and a late time that is long compared with the collision times. Similarly, vx 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 small-scale 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 vx and Te with and without collisions. With full collisions, fluctuations of electron density ne and temperature Te 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 vx value ∼5% (for the N7+ 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 Te 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 Te. The impact of thermal-effects-induced 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 Doppler-shift 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 up-shifted frequencies in the left half of the speckles and down-shifted frequencies in the right half of the speckles. At the laser beam scale, this leads to up-shifted frequencies in the left half of the laser beam and down-shifted frequencies in the right half of the laser beam. In the presence of ion trapping, such a Doppler-shift away from resonance can be comparable to that caused by trapping-modified 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.

CBET dynamics can be influenced by ion trapping and ion-heating-induced detuning,21 as well as by flow induced by thermal effects as explained above. Collisions may affect both processes: Collisional de-trapping is important near the onset threshold intensity for trapping; collisions also introduce thermal effects, enhancing the plasma flow. Since ion-electron collision rates are small compared with Ion–ion collision rates (see Sec. II C), only ion–ion collisions were included in prior CBET simulations21 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 ion-trapping and flow modifications in CBET.

In this work, the CBET gain percentage of the seed beam is defined as gain ( t ) = 100 % × [ P s out ( t ) P s in ( t ) ] / P s in ( t ), where P s in ( t ) and P s 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 ( P s out / P s in ) as used in linear theory.

FSRS may be unstable in the CBET amplified seed beam, as found in VPIC simulations relevant to indirect-drive.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.

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 × 10 13 W/cm2 (red curves) and high seed intensity I s = 5.0 × 10 14 W/cm2 (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.

FIG. 4.

CBET gain vs time from 2D simulations at a pump intensity I p = 2.2 × 10 15 W/cm2, and seed intensities I s = 1.0 × 10 13 W/cm2 (red curves) and I s = 5.0 × 10 14 W/cm2 (black, blue, and green curves). The solid, dashed, and dotted curves are from collisionless, Ion–ion collision, and full collision simulations, respectively.

FIG. 4.

CBET gain vs time from 2D simulations at a pump intensity I p = 2.2 × 10 15 W/cm2, and seed intensities I s = 1.0 × 10 13 W/cm2 (red curves) and I s = 5.0 × 10 14 W/cm2 (black, blue, and green curves). The solid, dashed, and dotted curves are from collisionless, Ion–ion collision, and full collision simulations, respectively.

Close modal

Differences in the CBET gain can arise from the following: (1) Full collisions lead to changes in ne and Te and modify the plasma conditions for CBET resonance. An increase in Te can also lead to a reduction of IAW damping. (2) Full collisions enhance laser-induced flow, as shown in Fig. 3, which also modifies the CBET resonance. (3) Collisional de-trapping 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 de-trapping of ions. In Fig. 5, at low seed intensity I s = 1.0 × 10 13 W/cm2, 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 de-trapping 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 N7+ 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 de-trapping alone since the ponderomotive-effects-induced 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.

FIG. 5.

2D simulation results with a pump intensity I p = 2.2 × 10 15 W/cm2 and a low seed intensity I s = 1.0 × 10 13 W/cm2, comparing effects of collisions on ion trapping shown by the ion 2D velocity space distributions f ( v x , v z ) at t = 90.0 ps (vx and vz are normalized to the ion thermal speed v th , i for each species). The characteristic distributions are shown near the center of the beam crossing region at x = 69.0 μm and z = 0.0 μm and are spatially averaged over a 1 × 1 μm2 domain. Frames (a1) and (b1), (a2) and (b2), and (a3) and (b3) are from collisionless, ion–ion collision, and full-collision simulations with the CBET gain shown by the red solid curve, the red dashed curve, and the red dotted curve in Fig. 4, respectively.

FIG. 5.

2D simulation results with a pump intensity I p = 2.2 × 10 15 W/cm2 and a low seed intensity I s = 1.0 × 10 13 W/cm2, comparing effects of collisions on ion trapping shown by the ion 2D velocity space distributions f ( v x , v z ) at t = 90.0 ps (vx and vz are normalized to the ion thermal speed v th , i for each species). The characteristic distributions are shown near the center of the beam crossing region at x = 69.0 μm and z = 0.0 μm and are spatially averaged over a 1 × 1 μm2 domain. Frames (a1) and (b1), (a2) and (b2), and (a3) and (b3) are from collisionless, ion–ion collision, and full-collision simulations with the CBET gain shown by the red solid curve, the red dashed curve, and the red dotted curve in Fig. 4, respectively.

Close modal

In Fig. 6, at low seed intensity I s = 1.0 × 10 13 W/cm2, 2D simulation results for flow velocity vx 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, laser-driven 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 vx and temperature along x (at z = 0.0 μm) for electrons (black curves), H+ (blue) and N7+ (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 ponderomotive-effects-induced flow is small and the ion velocity v x ( x ) is predominantly from the modification of the velocity space distribution due to ion-trapping. The effects of de-trapping 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 de-trapping, the amplitudes of vx and ion temperatures are lower in the ion–ion collision case.

FIG. 6.

2D simulation results at a pump intensity I p = 2.2 × 10 15 W/cm2 and a low seed intensity I s = 1.0 × 10 13 W/cm2, comparing the effects of collisions on flow velocity vx (normalized to the IAW phase velocity v ph) and temperature (normalized to the initial temperature for each species). The profiles along x at times t = 50.0 ps and 91.5 ps for electrons (black curves), H+ (blue) and N7+ (red) ions are averaged over 20 cells around z = 0.0 μm. Frames (a1)–(d1), (a2)–(d2), and (a3)–(d3) are from collisionless, ion–ion collision, and full-collision simulations with the CBET gain shown by the red solid curve, the red dashed curve, and the red dotted curve in Fig. 4, respectively.

FIG. 6.

2D simulation results at a pump intensity I p = 2.2 × 10 15 W/cm2 and a low seed intensity I s = 1.0 × 10 13 W/cm2, comparing the effects of collisions on flow velocity vx (normalized to the IAW phase velocity v ph) and temperature (normalized to the initial temperature for each species). The profiles along x at times t = 50.0 ps and 91.5 ps for electrons (black curves), H+ (blue) and N7+ (red) ions are averaged over 20 cells around z = 0.0 μm. Frames (a1)–(d1), (a2)–(d2), and (a3)–(d3) are from collisionless, ion–ion collision, and full-collision simulations with the CBET gain shown by the red solid curve, the red dashed curve, and the red dotted curve in Fig. 4, respectively.

Close modal

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 N7+ 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 thermal-effects-induced flow in the pump beam, the profile of v x ( x ) is asymmetrical about the center of x-domain (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 slowing-down, perpendicular diffusion, and thermalizing processes, leading to a flow v x ( x ) resembling that in the pump-only 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 Te up to 15%.

At the high seed intensity I s = 5.0 × 10 14 W/cm2, in Fig. 7, 2D simulation results for flow velocity vx 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 de-trapping and flow. These results explain why the CBET gain at high seed intensity is insensitive to collisional de-trapping and thermal-effects-induced 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

FIG. 7.

2D simulation results at a pump intensity I p = 2.2 × 10 15 W/cm2 and a high seed intensity I s = 5.0 × 10 14 W/cm2, comparing effects of collisions on flow velocity vx (normalized to the IAW phase velocity v ph) and temperature (normalized to the initial temperature for each species). The profiles along x at times t = 50.0 ps for electrons (black curves), H+ (blue) and N7+ (red) ions are averaged over 20 cells around z = 0.0 μm. Frames (a1) and (b1) and (a2) and (b2), are from ion–ion collision and full-collision simulations with the CBET gain shown by the blue dashed curve and the black dotted curve in Fig. 4, respectively.

FIG. 7.

2D simulation results at a pump intensity I p = 2.2 × 10 15 W/cm2 and a high seed intensity I s = 5.0 × 10 14 W/cm2, comparing effects of collisions on flow velocity vx (normalized to the IAW phase velocity v ph) and temperature (normalized to the initial temperature for each species). The profiles along x at times t = 50.0 ps for electrons (black curves), H+ (blue) and N7+ (red) ions are averaged over 20 cells around z = 0.0 μm. Frames (a1) and (b1) and (a2) and (b2), are from ion–ion collision and full-collision simulations with the CBET gain shown by the blue dashed curve and the black dotted curve in Fig. 4, respectively.

Close modal

Note that for the 2D simulations, the average intensity for the speckled pump and seed beams is defined I ave I 2 / I , 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 I and using simple linear theory estimates,21 we find that the gain percentage from linear theory is 49% (using the interaction length L = d / sin θ 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.

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  × 17.3 × 54.0 μm3 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 × 17.3 μm2 in their respective launch planes. With more than 5 × 109 cells and 80 particles per cell, the simulations employ 211 600 MPI ranks and over 1.2 × 1012 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 λ 0 = 2.8 μm and length 2 π F 2 λ 0 = 99.0 μm, these simulations represent the multi-speckled nature of the 3D CBET interactions. The 3D geometry allows for additional laser-driven 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 × 10 13 W/cm2 and high seed intensity I s = 5.0 × 10 14 W/cm2 are shown in Fig. 8(a). (Note that here, in contrast with 2D simulations, we use the more traditional definition of intensity I s I , 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 de-trapping and thermal-effects-induced flow, as well as from changes in the plasma conditions in the presence of full collisions.

FIG. 8.

3D CBET simulations at a pump intensity I p = 2.2 × 10 15 W/cm2, with either a low seed intensity I s = 1.0 × 10 13 W/cm2 or a high seed intensity I s = 5.0 × 10 14 W/cm2. (a) CBET gain vs time without collisions (solid curves) and with full collisions (dotted curves) at low seed intensity (red curves) and at high seed intensity (black and green curves). For the 3D simulation with full collisions at low seed intensity, profiles along x (averaged over 10 cells around y = z = 0 at time t = 64.2 ps) for charge density (b), flow velocity vx (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 N7+ (red) ions. Similarly, for the 3D simulation with full collisions at high seed intensity, profiles along x for flow velocity vx (e) and temperature (f) are shown for the electrons (black curves), H+ (blue) and N7+ (red) ions.

FIG. 8.

3D CBET simulations at a pump intensity I p = 2.2 × 10 15 W/cm2, with either a low seed intensity I s = 1.0 × 10 13 W/cm2 or a high seed intensity I s = 5.0 × 10 14 W/cm2. (a) CBET gain vs time without collisions (solid curves) and with full collisions (dotted curves) at low seed intensity (red curves) and at high seed intensity (black and green curves). For the 3D simulation with full collisions at low seed intensity, profiles along x (averaged over 10 cells around y = z = 0 at time t = 64.2 ps) for charge density (b), flow velocity vx (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 N7+ (red) ions. Similarly, for the 3D simulation with full collisions at high seed intensity, profiles along x for flow velocity vx (e) and temperature (f) are shown for the electrons (black curves), H+ (blue) and N7+ (red) ions.

Close modal

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 vx (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 N7+ (red) ions. Thermal effects in the intense pump beam are stronger in 3D, as evidenced by the stronger heating of Te in 3D with an increase in Te 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 x-domain (x = 35 μm) reflects the presence of induced flow superimposed onto an all-positive 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 vx (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 side-loss ions as compared with 2D. The side-loss ions originate from the trapped ions that escape the narrow IAW potential well63 at a rate v th , i / F λ 0. The side-loss 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 side-loss 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 side-loss 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 wave-particle interactions and electrons heated by the laser can propagate out of the boundaries and are replenished by Maxwellian and super-Gaussian 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 laser-driven flow and temperature fluctuations in the y dimension, but the boundary conditions are periodic in y. Ions accelerated by wave-particle 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 

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 laser-driven flow and temperature fluctuations have been examined. In the mid-Z plasma with an intense pump laser beam, ion trapping, collisional de-trapping, 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 de-trapping 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 de-trapping 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 × 10 14 W/cm2. 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 low-intensity 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 × 10 15 W/cm2) 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 pump-only simulation with full collisions at a lower pump intensity I p = 1.0 × 10 15 W/cm2 has also been performed and the plasma flow vx profiles are shown in Fig. 3(d) by the orange and green curves for N7+ and H+ ions, respectively, with the peak vx values ∼4% to 5% of the IAW phase velocity v ph (compared to the ∼5% to 10% at I p = 2.2 × 10 15 W/cm2 that can Doppler-shift 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 × 10 15 W/cm2. For these simulations, the average intensity for the speckled beam defined as I ave I 2 / I are used. Converting the intensity to that based on the normal average intensity definition I , the pump beam intensities for these simulations are I p = 1.32 × 10 15 and 6 × 10 14 W/cm2.

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 Thomson-scattering system measurements.19 Together with the P 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 fusion-relevant laser-plasma configurations as discussed in prior work.26,27 Collisional de-trapping 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 direct-drive ICF implosion. Also, in the low plasma density regime, as in the case for low gas fill and near-vacuum 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 λ 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 λ 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 direct-drive 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 density64,65 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, thermal-effects-induced flow would be expected to have similar detuning effects on CBET, a point we intend to consider in future work.

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. DE-NA0003856. 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).

The authors have no conflicts to disclose.

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).

The data that support the findings of this study are available within the article.

To model the OMEGA CBET experiments, our VPIC calculations require that we load non-Maxwellian (super-Gaussian) electron velocity distribution functions initially and that the particle-refluxing boundary conditions reinject these distributions at the boundary. We use a distribution function of the form given by Eq. (1). For notational convenience, we define u v / v m, so that
f m ( u ) d 3 u = C m v m 3 exp ( u m ) d 3 u .
(A1)
We will be using the rejection sampling method so we omit normalization constants C m v m 3 in our discussion of the sampling. (The constants can be absorbed into the definitions of the reference functions.) We consider here the special case m = 3 though this technique could in principle be modified to apply to any m.
For the initial VPIC particle load, we wish to sample from the distribution f 3 ( u ). This is done using the rejection sampling method, noting that for all positive u, u 2 f 3 ( u ) f env ( u ), where
f env u exp ( u 2 )
(A2)
is an envelope function that is easily inverted. We sample elements of the distribution f3 through the following steps:
  1. Generate four samples p1, p2, p3, and p4 independently from a uniform random distribution from zero to one.

  2. Pick a sample speed u samp = log p 1, obtained from inverting f env ( u ).

  3. If p 2 f env ( u samp ) u samp 2 f 3 ( u samp ), accept the sample and proceed to step 4. Otherwise, reject the sample and return to step 1.

  4. Generate the components of u by sampling uniformly over solid angle
    μ = 2 ( p 3 1 2 ) ϕ = 2 π p 4 u x = u samp cos ϕ 1 μ 2 u y = u samp sin ϕ 1 μ 2 u z = u samp μ .
This prescription is quite efficient, as seen by computing the rejection probability
0 d u [ f env ( u ) u 2 f 3 ( u ) ] 0 d u f env ( u ) = 0.167 .
(A3)
In other words, only one in six samples are rejected on average.
For the VPIC reflux boundary handler, we require separate sampling of normal and tangential velocities through the use of a flux boundary condition, i.e., normal velocity sampled from u z f 3 , z ( u z ), where uz is the velocity component normal to the boundary. We get f 3 , z ( u z ) by integration over perpendicular velocity components
f 3 , z ( u z ) d u z = d u z 0 d 2 u f ( u ) = 2 π d u z 0 d u u exp [ ( u 2 + u z 2 ) 3 / 2 ] = 2 π 3 Γ ( 2 3 , u 3 ) ,
(A4)
where
Γ ( a , z ) = z d t t a 1 e t
(A5)
is the incomplete Gamma function.72 To within 0.3% error over all non-negative uz, u z f 3 , z ( u z ) is fit by
u z f 3 , z fit ( u z ) = u z exp ( n = 0 6 c n u z n ) ,
(A6)
where
c 0 = + 1.042 809 c 1 = 0.013 657 0 c 2 = 1.030 15 c 3 = 0.190 343 c 4 = 0.318 559 c 5 = + 0.065 128 1 c 6 = 0.005 467 27
and u z f 3 , z ( u z ) < 2.83 f env ( u z ) for all non-negative uz, with f env ( u z ) defined as above. Therefore, we may use a rejection sampling method as before to sample from u z f 3 , z ( u z ):
  1. Generate two samples p1 and p2 independently from a uniform random distribution from zero to one.

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

  3. 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.

Once we have a sample of the normal velocity, we need to sample the velocity tangential to the boundary f 3 , ( u | u z ) d 2 u , the (conditional) probability that the perpendicular velocity falls within a perpendicular-velocity-space areal element d 2 u located at u if the speed normal to the boundary is uz. Owing to symmetry,
f 3 , ( u | u z ) d 2 u = 2 π u f 3 , ( u | u z ) d u .
(A7)
The distribution function 2 π u f 3 , ( u | u z ) 1.165 ( 2 π ) e u z 3 f env ( u ), where f env is defined as above. Therefore, we can again apply the rejection sampling method:
  1. Generate three samples p1, p2, and p3 independently from a uniform random distribution from zero to one.

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

  3. If 1.165 exp ( u z 3 ) p 2 f env ( u samp ) > u samp exp [ ( u z 2 + u samp 2 ) 3 / 2 ], reject the sample and return to 1.

  4. Set ϕ = 2 π p 3 and set the two tangential components u t 1 , u t 2 to
    u t 1 = u samp cos ϕ u t 2 = u samp sin ϕ .
    (A8)

This sampling rejects a fraction 0.41 of trials on average. For numerical stability, it is recommended that uz be rejected above uz = 4 prior to entering this portion of the sampling algorithm. Overall, for VPIC calculations on a variety of many-core supercomputers available to us, the Maxwellian reinjection of super-Gaussian distributions adds of order 10%–20% overhead to the runtime, making these runs practicable.

In setting up our simulations, we need to obtain frequency and wavenumber matching conditions, which requires that we solve the dispersion relation ε ( ω , k ) = 1 + χ e ( ω , k ) + χ i ( ω , k ) = 0 where the electron susceptibility χe is evaluated with an m = 3 super-Gaussian and χi is given in terms of the plasma dispersion function49 in the standard way. For the special case m = 3, the one-dimensional distribution function f ( v ) along IAW wavevector k is
f ( v ) = 2 π 0 d v v C 3 exp [ ( v 2 + v 2 v 3 2 ) 3 / 2 ] ,
(A9)
with v3 defined as above. After a bit of algebra, it can be shown that
f ( v ) = 2 π 3 C 3 v 3 2 Γ [ 2 3 , ( v v 3 ) 3 / 2 ] .
(A10)
The electron susceptibility is
χ e ( ω , k ) = ω pe 2 k 2 d v f ( v ) / d v v ( ω / k ) ,
(A11)
where to maintain causality the integration contour is understood to pass below the pole in the denominator. This expression can be evaluated for the m = 3 super-Gaussian and shown to be of the form
χ e ( ω , k ) = [ Γ ( 5 3 ) ] 3 / 2 2 3 v th , e 3 ω pe 2 k 2 d v v exp [ ( v 2 / v 3 2 ) 3 / 2 ] v ( ω / k ) ,
(A12)
which we evaluate numerically in root-finding software such as Mathematica to obtain roots of the dispersion relation.
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