Wavelength-detuning cross-beam energy transfer mitigation scheme for direct drive: Modeling and evidence from National Ignition Facility implosions

In direct-drive inertial confinement fusion (ICF), laser beams irradiate a low-Z shell (plastic or Be) containing a layer of frozen deuterium–tritium (DT) and ablatively drive an implosion. The ultimate goal of ICF is ignition and energy gain; the minimum shell kinetic energy required for ignition (defined as when the energy from DT fusion reactions exceeds the laser energy incident on the target) is given by (Ref. 1)

where the three implosion parameters—$α,vimp,$ and P_abl [adiabat (the ratio of the fuel pressure to the Fermi-degenerate pressure at peak implosion velocity), implosion velocity, and ablation pressure, respectively]—are determined by the deposition of the laser energy into the coronal plasma of the target, the subsequent heat conduction to the ablation surface, and the resulting equation of state (EOS) of the shell material. Cross-beam energy transfer (CBET)² has been identified in direct-drive experiments on the OMEGA³ and National Ignition Facility (NIF)⁴ Laser Systems to reduce absorption and thereby ablation pressure and implosion velocity. The CBET effect reduces the laser absorption by 20% to 30%, the implosion velocity by 10%–15% and the ablation pressure by up to 50% in ignition-scale designs. Reducing the target mass compensates for CBET losses, but the thinner shells become compromised as a result of hydrodynamic instability growth.⁵ As Eq. (1) indicates, the significant loss caused by CBET precludes ignition when E_min exceeds the laser system's capabilities, motivating CBET mitigation as a critical challenge facing direct drive.

The role of CBET in direct drive was identified in early research^6,7 but only recently identified as the leading cause of decreased energy coupling. The critical role of CBET became apparent when attempts were made to match multiple calculated observables (shell morphology, trajectory, scattered-light spectra and power, and shock timing) with experiments.^8,9 Historically, the role of CBET was masked by using a flux-limited electron-transport model,⁴⁵ reproducing either scattered-light power or shell trajectory but ultimately unable to match other observables. Good agreement with all of the multiple experimental observables occurred^8,9 with the inclusion of both the CBET and nonlocal electron transport¹⁰ models in 1-D LILAC¹¹ and 2-D DRACO¹² simulations.

CBET laser–plasma interaction (LPI) results from two-beam energy exchange via stimulated Brillouin scattering (SBS),² which reduces absorbed light and consequently reduces ablation pressure and implosion velocity. The dominant CBET loss mechanism in direct drive occurs when rays counter-propagate (backscatter mode), thereby increasing scattered light eliding absorption, as illustrated in Fig. 1(a). For the ignition-relevant overlapped peak-integrated beam intensities of ∼4–8 × 10¹⁴ W/cm² for the NIF experiments described in this paper, CBET is calculated to reduce laser absorption by 22%, the average implosion speed by ∼9%, and the average ablation pressure by 35%. While these numbers are lower than expected for ignition-scale designs with similar overlapped intensity, the coronal plasma volume is smaller, corresponding to the appropriately scaled target radius and shot energy available for these experiments that was determined by using the existing far-field spot sizes currently equipped on the NIF beams. These drive-related results are consistent with other ongoing OMEGA-⁸ and NIF-scale⁹ experiments.

Mitigation strategies for the deleterious CBET effects invoke combinations of spatial (by reducing the interacting beam size and potentially on-target uniformity, beam overlap, and speckle reduction),^13,14 temporal (via altering the timing of interacting beams),¹⁵ and wavelength domains.¹⁴ Wavelength detuning, the focus of this paper, works by altering the resonance condition between interacting beams.² Utilizing wavelength separation and continuous, wide bandwidth to defeat laser–plasma interactions (LPI's) dates back many decades with a rich history.^16–18 Distributing the laser energy into separated or continuous-wavelength bands decreases the intensity of the individual interactions, shifts resonance attributes, and potentially disrupts the coherent buildup within the plasma with a sufficiently wide bandwidth for each LPI process. The seeded SBS process of CBET does not result from noise like other LPI processes, but rather from another coherent beam, and its suppression requires larger bandwidths since the growth time-scales are shorter. In addition, even with a low gain coefficient, CBET levels can be large because of the large initial amplitude of the “seed.” Wavelength detuning was first examined for indirect drive,¹⁹ and subsequently for direct drive, but prematurely dismissed as a viable option.²⁰ 

After a more-detailed discussion of the CBET modeling process used in DRACO, the wavelength-detuning mitigation scheme will be presented using experimental evidence from the NIF Laser System. The amount of wavelength detuning currently available on the NIF (±2.3 Å UV) sufficiently demonstrates CBET mitigation for these initial proof-of-principle experiments. Wavelength detuning combined with a unique spatial approach will be discussed in the section on the OMEGA Laser System that applies to any direct-drive ICF laser system. The spatial approach employs a technique applied to a distributed phase plate (DPP) referred to as spot-masking apodization (SMA),²¹ which enhances CBET mitigation while having low impact on target illumination uniformity, beam overlap, or speckle reduction.

In direct drive, many overlapping beams interact with each other in a complicated tangle of intensity, directions, and wavelengths depending on the beam port, laser, and pulse-shape configurations surrounding the imploding target. In addition, each beam strongly refracts and frequency chirps in the expanding, evolving plasma atmosphere during propagation and then scatters energy spectra in a wide spread of exiting paths. The radiation–hydrodynamics code DRACO is an arbitrary Lagrangian–Eulerian (ALE) simulation code capable of modeling ICF implosions, including the effects of CBET in a 2-D polar-cylindrical mesh. The DRACO CBET package (Adaawam²²) is an integral part of the highly scalable parallel 3-D ray-trace package (Mazinisin²³),²⁴ which models each beam as a set of adaptively chosen rays to minimize noise; see  Appendix A for more details. The Mazinisin package propagates refracting rays in 3-D within a 2-D plasma atmosphere and accounts for arbitrary multiple beam orientations, arbitrary far-field spot-shapes, independent pulse shapes, beam repointing, target offset, laser absorption, frequency chirp, speckle imprint, and smoothing by spectral dispersion (SSD)²⁶ and for many nonuniform and random beam-to-beam variations such as pulse power imbalances, mistiming, or mispointing, as well as the 3-D CBET interaction effects (using Adaawam) with or without speckled beams. DRACO is able to emulate the effect of 3D modal power induced by laser imbalances by imposing any non-zero m-mode spectral power into a corresponding m = 0,ℓ-mode perturbation. Since the radiation-hydrodynamics code DRACO is only 2-D, the 3-D spatial feedback into the corona cannot be modeled; however, some useful 3-D scattering and deposition effects of refraction and CBET are captured within diagnostic output and can be correlated to experiment. This section concentrates on the modeling of CBET.

An extension to the plane-wave CBET model² (see Fig. 2) adapts the steady-state and strong-damping limit fluid model to 3-D interacting rays in Adaawam by generalizing the wave vector description, accounting for the probe (k_probe) and pump (k_pump) rays' direction and wavelength and their connection to the ion-acoustic wave vector description, viz., $ka≡ kpump −kprobe$⁠. The extension expressed as an exponential factor, dτ_CBET, describes the amount of energy either gained or lost via an interaction with a background pump field over a small path length, viz.,

where

and where the spatially dependent plasma variables, $⟨Z⟩,$T_e, T_i, v_fluid, and c_a are the average atomic number, the electron and ion temperatures, the expanding plasma fluid velocity, and the sound speed, respectively; ν_a is the dimensionless ion-acoustic damping coefficient [set to 0.2 using the 1-D code LILAC as guidance (Ref. 20)]; the ray propagation–dependent variables ds, ω_probe, and ω_pump are the probe ray's path length and the probe and pump angular frequencies, respectively; in addition, the physical constants e, m_e are the electron charge and mass and the basic light constants c and λ₀ are the vacuum speed of light and vacuum central laser wavelength, respectively. The normalized electron number density is defined as $ne′≡ne/necrit ,$ where n_e is the spatially dependent electron number density and the critical electron number density is given by $necrit≡meω02/(4πe2),$⁴⁴ where $ω0≡2πc/λ0$ is the central laser angular frequency. The resonance function P(η), defined in Eq. (3), determines the relative strength of the CBET interaction and gain or loss depending on its sign because all other parameters in Eq. (2) are strictly positive. The value and sign of the resonance function depend directly on the wave-vector–matching condition η given in Eq. (4), where gain occurs when $η>0$ and loss for $η < 0.$ Perfect wave-vector matching occurs when $η=±1,$ which directly corresponds to the maximal amplitude $P|η=±1=±1/νa$ at resonance. Random polarization is accounted for using Eq. (5), but this relationship is not exactly known and is a potential reason for the need of the experimental adjustment parameter f_exp that is lumped into that expression. Further research into the polarization dependence is expected to help determine the proper formulation of ς_pol and possibly reduce the required value of f_exp. A tunable beam with variable polarization is presently being added to the OMEGA laser for fundamental research on CBET.²⁵ In addition, since the factor ν_a proportionally changes the resonance function as $P|η=±1=±1/νa$ the value used could account for a portion of the experimental adjustment factor f_exp. Future investigation of this constant will be pursued to assess the correct value given the potential of wavelength or hydrodynamic dependence of the value of ν_a.

Rays transport energy and not intensity; for this reason, modeling the CBET interaction [as well as inverse bremsstrahlung (IBS)] involves differential changes of the probe ray's energy. The only two LPI effects simulated in DRACO are IBS and CBET. The resultant probe-ray energy, $Eprobe1$ for any single CBET interaction with a particular pump-intensity–field, wave-vector pair ${Ipump,kpump}$ over a small path length ds is given by $Eprobe1=Eprobe0 exp (dτCBET),$ where $Eprobe0$ is the probe ray's energy prior to the interaction. The probe ray's energy gain (when dτ_CBET > 0) or loss (when dτ_CBET < 0) resulting from the single interaction is calculated by $ΔEprobe=Eprobe0[exp (dτCBET)−1].$ Notice that the exponential gain/loss factor $dτCBET$ is dependent only on the pump intensity in this formulation and not on the probe ray's intensity or energy. Therefore, any swelling of the probe-ray intensity that might occur during refraction in small volumes after a turning point cannot affect the outcome of the probe ray's energy gain/loss. However, a significant increase in intensity directly corresponds to a proportionally increased ray-density which results in more individual CBET interactions which may result in increased net CBET gain and loss if the interaction volume is significant.

The CBET model given by Eqs. (2)–(5) includes relevant SBS physics and results in gain/loss for a probe ray interacting with the total background pump field as the probe refracts and traverses the computational mesh. This modeling assumes that the CBET response is steady-state based on the strong damping limit which is qualified by the 10ps stabilization time of detailed full-field simulations⁴⁷ which is faster than typical 100ps hydrodynamic responses. The CBET modeling presented in this paper assumes there is a linear response in that there is no dependence on the probe intensity which is valid for the intensity ranges presented here. DRACO does have the facility to model the non-linear response for cases of high intensity fields which allows for the saturation effect of the electron density in relation to the IAW where the CBET response will then depend on both the intensity of the pump and probe. It is curious to note that including this dependence on the probe intensity actually helps stabilize the CBET interactions by making the formulation symmetric; however, this modeling then adds another adjustable parameter that dictates the saturation limit. DRACO also has options (with and without saturation) to run using a kinetic based model. These kinetic based options were not fully developed based upon the observations discussed in Ref. 20 given the intensities and scale-lengths of current experiments; the 1-D modeling was used to guide the development effort of DRACO for the sake of time. If the fluid model falls short of emulating experiments DRACO is prepared for the task of simulating with the kinetic models.

The background pump-intensity field I_pump represents an accumulation of all rays from any port traversing each computational zone while accounting for all their intensities, wave vectors, and chirped wavelengths in an angular spectrum representation (ASR)^26,27 that divides the accumulated ray energies (weighted as multiplying by path length and dividing by volume, viz., the volume-weighted intensity) into angular and wavelength spectral components (see Fig. 3). In a typical simulation, 50–75k rays commonly traverse cells of dimension 20–50um (imprint simulation uses finer resolution but similar ray density) and accumulate to form the ASR in the active CBET region. The volume weighted average is the appropriate quantity corresponding to the inherent hydrodynamic scale of interest in the simulation; if smaller spatial scales are important then the mesh is designed to have higher resolution. An irregular geometry of a cell is properly handled using this volume weighted intensity construction. For example, a wide cell-area accepts proportionally higher ray density while the shorter cell-depth along the propagation direction governs the pathlength. The opposite aspect ratio yields a smaller ray density with longer pathlengths. In addition, rays from the same port average in intensity whereas rays from differing ports add in intensity; this could not be handled correctly if rays tracked intensity in lieu of energy.

Each computational zone in the corona contains a unique ASR divided over all propagation angles and wavelengths that defines the background pump field illuminating the zone. For example, the total overlapped inbound intensity of the pump field may be ∼1 × 10¹⁵ W/cm² for all beam ports incident on a cell but this summation represents an ASR that is distributed over many different directions and wavelengths, effectively lowering the intensity of each pump-field interaction and where the net CBET response is obviously the sum over all the individual interactions which are individually dependent on intensity, angles and wavelength. The calculated pump-field intensity accounts for all relevant changes caused by IBS absorption, gain/loss as a result of CBET, and any effective increase or decrease caused by refraction. In fact, the potential intensity increase from refraction located after the turning points (defined as the minimum radius reached where the direction vector becomes normal to the position vector) are inherently included and define a refractive shadow boundary surface with a sharp interface. The intensity increase from refraction can become significant (but is never infinite because rays never cross each other at the same optical pathlength, i.e., always non-zero ray-area, under relevant ICF conditions which only generates a weak form of volume focusing in stark contrast to important distributed focusing like from lens aberrations that occupy significant volume and families of rays converging into the same region); this increase tends to be balanced by occurring within vanishingly small volumes where the net effect is properly handled via the volume weighted intensity. This volume tends to be small because, away from the refractive shadow boundary, refraction in a negative electron density gradient profile is highly divergent which restricts the intensity peaks to small volumes. Any corresponding increase in CBET gain or loss is inherently handled in DRACO since an increase in intensity directly relates to a proportional increase in ray density which increases the density of individual CBET interactions; the net result may increase CBET gain and loss only if the interaction volume is significant.

Any probe ray that traverses a computational zone experiences many instances of CBET gain and loss via summations over all interactions with the entire ASR as well as losses via collisional IBS. All instances of CBET and IBS are accumulated into a total exponential gain/loss factor $dτ=dτCBETgain+dτCBETloss+dτIBS$ over the ray's trajectory through the computational zone. The ray's trajectory is not directly affected by CBET or IBS, but solely determined by the coronal electron density profile of the current time step. However, the trajectory eventually becomes altered by the combined effects of CBET and IBS indirectly since the increased gain and/or scatter and collisional losses eventually change the coronal temperature gradients, which ultimately feed back into the electron density gradients during the hydrodynamic evolution. The total exponential factor dτ represents an accumulation of all gains and losses from the entire ASR and IBS along the trajectory within a computational cell. The total change of a probe ray's energy is given by $ΔEprobe=Eprobe0[exp (dτ)−1],$ which asymptotically goes to $ΔEprobe ∼Eprobe0dτ(1+0.5dτ)$ as $dτ →0.$ Additionally, the ray's average energy over the path length, ds, is exactly given by $⟨Eprobe⟩=ΔEprobe/dτ,$ which asymptotically goes to $⟨Eprobe⟩ ∼Eprobe0(1+0.5dτ)$ as $dτ →0.$ However, after all the gain/loss effects accumulate into the total dτ, subsequent separation into different gain and loss energy components become intractable, in general, because of the nonlinear nature of the of the exponential operator. It is tempting to assert that, for example, the total CBET gain, $ΔEprobegain$ is of the form $Eprobe0[exp (dτCBETgain)−1],$ where $dτCBETgain$ is the total accumulated CBET gain portion. This is mathematically incorrect because the exponential operators are not separable, resulting in different answers depending on the order of operations taken to separate gain from loss; attempting to balance this disparity using an average of gain/loss and loss/gain is equally unsound and computationally inefficient. If the ray's path length, ds, is small enough such that each elemental contribution to the total dτ is roughly constant or the energy changes are evenly distributed over the whole path, it can be shown that gains and losses separate via a simple ratio scheme. This assumption is consistent with the usage of an ASR that represents the entire computational cell. Assuming constant values or an even distribution of the dτ components over the path length ds total gain and loss components are exactly separable within a sufficiently small volume, e.g., the computational cell, using a straightforward ratio scheme. The total CBET gain, CBET loss, and IBS loss are given by $ΔEprobegain=dτCBETgain$ $⟨Eprobe⟩, ΔEprobeloss=dτCBETloss⟨Eprobe⟩,$ and $ΔEprobeIBS=dτIBS⟨Eprobe⟩,$ respectively, which are exact and computationally efficient under the constant cell-wise or evenly distributed ASR/IBS assumption.

Adaawam calculates the CBET interaction self-consistently in conjunction with the hydrodynamic evolution of the ICF target (via a split-step technique). In addition, Adaawam captures all the necessary coupled interactions between the dynamic electron density profile, temperature, and plasma-flow velocity that dictates the behavior of CBET and vice versa resulting from the strongly coupled CBET and hydrodynamics, which significantly affect each other. Averaging out coronal angular spatial variations of the plasma before calculating CBET or employing a corona not derived using CBET (or from a lesser dimensional code) and its induced spatial variations completely misses the coupling of this essential spatial effect. Adaawam uses advanced iterative feedback control [proportional-integral-differential (PID) control loops²⁸] to stabilize the CBET coupled many-beam interactions while maintaining energy conservation (see Fig. 4). The DRACO CBET model compares with many experimental observables across a range of implosions on OMEGA²⁹ and the NIF.⁷ An experimentally determined CBET gain multiplier of f_exp = 1.5 (from unrelated OMEGA shots²⁹) that uses the first-principles EOS tables was applied to all pre- and post-shot simulations without attempting to fit the NIF shots having similar intensity but different scale lengths and pulse shapes. The CBET gain multiplier of f_exp = 1.5 that applies across laser systems indicates a predictive ability in regards to shell morphology and trajectory on the initial wavelength-detuning shot campaign at the tested ∼4–8 × 10¹⁴ W/cm² intensity range. The nonlocal electron transport package¹⁰ is used in modeling experiments on both OMEGA and the NIF.

Maximal CBET occurs in the rapidly expanding coronal plasma where two interacting rays satisfy the ion-acoustic-wave–matching conditions [i.e., Eq. (4), when η = ±1] that account for propagation direction, wavelength, and fluid flow; e.g., a CBET resonance occurs at the Mach-1 surface given a radial plasma flow for directly opposed radially propagating rays of equal angular frequency or wavelength. The significant CBET-interaction region for a directly driven imploding ICF target tends to form at supersonic speeds above the Mach-1 surface for laser beams without detuning (e.g., 1 ≲ Mach ≲ 1.5). The ray's angular frequency is given by its initial value and the integrated temporal derivative of the refractive index (a generalization of the conventional Doppler shift³⁰). The expanding plasma dynamically alters the instantaneous refractive index in space via the temporal dependence electron density, and thereby the angular frequency and wavelength, and is independent of ray propagation direction¹⁴ (see the  Appendix B for more details). Consequently, the CBET resonance features adapt as the coronal plasma evolves along with the interacting ray wavelengths. These resultant interactions directly map onto a frequency-chirped scattered-light measurement that can be employed to analyze the implosions and laser–plasma interaction physics. The 2-D code DRACO produces chirped scattered-light diagnostics that have azimuthally varying features resulting from the variation in the modeled 3-D beam-port geometry coupled with the polar and azimuthal angular dependence of CBET (see Figs. 5 and 6). However, as self-emission measurements (see Fig. 7) indicate, there are additional azimuthal variations in the ablation surface (not included in DRACO since the necessary azimuthal feedback cannot be modeled in a 2-D code) that are imprinted in the corona; these azimuthal gradients consequently alter the energy deposition and scattered light. Many closely spaced experimental diagnostic ports are necessary to correlate them on average to a 2-D simulation, which does not include the necessary 3-D feedback into the electron density to reproduce an experiment without averaging (as previously addressed). To resolve these expected variations, future 3-D simulations will attempt to verify the measured azimuthal variations and determine the magnitude of this effect, the impact on the implosions, and possibly alter the pointings to minimize the azimuthal modulations. Alternatively, increasing the number and distribution of scattered-light diagnostic parts could help resolve any discrepancies observed.

Wavelength detuning between crossing beams responds differently in indirect- versus direct-drive ICF implosions, depending on the dominant CBET mode. In indirect drive, the sign of small wavelength detuning (<2 Å UV) is used to control the direction of energy transfer between co-propagating interacting beams by leveraging the CBET resonance for the forward-scatter (or sidescatter) mode where the pump and probe beams are either both inbound or outbound,¹⁴ i.e., $|ka⋅ vfluid|≪|ωpump−ωprobe|$ in Eq. (4) (see Fig. 8 for an illustrated example), implying that the frequency difference controls the sign of $η$⁠. The magnitude of the $ka⋅ vfluid$ term becomes insignificant for two reasons: (1) the pump- and probe-vectors co-propagate within a small angle, leading to a small-magnitude ion-acoustic wave vector, $ka≡ kpump −kprobe;$ and (2) its direction is nearly orthogonal to the fluid velocity, v_fluid. The main CBET interaction occurs near the laser entrance hole of the indirect-drive hohlraum. The forward-scatter mode effect on spatial distribution of laser-energy deposition is more dramatic in indirect drive because there is significant propagation (or throw) after the interaction. The propagation distance separates the far-field spots before deposition on the inside of the hohlraum wall. Any forward-scatter mode interactions that occur change the balance on the hohlraum wall because of the large spot separation. While this mode occurs in direct drive, it negligibly alters scattered-light loss because any energy exchanged is merely deposited in slightly different regions (because of the short propagation distance before deposition and the significant beam overlap) contributing to distortion; some modest additional distortion at small wavelength separations can arise.¹⁴ In contrast, an outbound ray in the dominant backscatter mode in direct drive always experiences CBET gain regardless of the wavelength-difference sign or magnitude (for nominal levels) because the ion-acoustic wave's contribution dominates the CBET resonance function, i.e., $|ka⋅vfluid|≫|ωpump−ωprobe|$ in Eq. (4) (see Fig. 2 for an illustrated example), implying that the frequency difference cannot control the sign of $η$ under normal circumstances.¹⁴ Under atypical conditions, the outbound ray may experience a loss resonance for extreme wavelength separation (e.g., >20 Å UV) but insignificantly impacts scattered light because the outbound rays typically transport little energy.

The ensemble CBET exchange is best described as an interaction volume (a weighted volume that determines the interaction strength, which depends on path length, intensity, wavelength, electron density, coronal temperature, fluid velocity, etc.) because any high-gain region is equally matched by loss and significant CBET occurs only when the ensemble interaction volume is large. For example, there might be high intensity after a turning point over insignificant path lengths and a diminishing area that form an ineffective and vanishingly small interaction volume with minimal resulting CBET. In addition, the physical diffraction-limited transverse speckle sizes formed from realistic focused beams are significantly larger in volume, which disrupt and diffuse any small-scale (∼micron or less) diffraction effects from self-interfering idealized smooth beams near the refractive shadow boundary. For speckle or the diffraction pattern to cause a CBET response to deviate significantly from this linear view the conditions untypical or irrelevant to ICF implosions need to exist to induce nonlinear behavior which also imparts saturation effects. As examples, this can occur using a construction of speckle not adhering to the known Goodman spectrum²¹ resulting in incorrect amplitudes or widths, not modeling refraction in spherical geometry causing speckle to interact over long distances by minimizing transverse movement of speckle and artificially enhancing the CBET response or combining all the power into a single simulated beam (or too few beams) instead of from many beams (>10 for OMEGA or >40 for NIF) normally overlapping from a wide distribution of beam angles resulting in non-ICF relevant conditions. There is little evidence to suggest that nonlinear conditions occur in typical direct-drive ICF implosions but they may become important to some degree at ignition scale. To be hydrodynamically significant, the average energy of rays over the entire computational cell must be significant, i.e., the intensity defined as energy times the path length divided by cell volume should be substantial. This definition is consistent with the CBET interaction via the ASR that represents the entire computational cell. A method that views the CBET interaction over the entire cell but employs high intensities that arise from refraction without compensating for the proportionally diminishing cross-sectional area and very small path lengths will inadvertently overpredict the CBET response. In contrast, attempting to model field-swelling by applying a maxima function actually dampens or saturates the CBET response rather than modeling what the name suggests, i.e., swelling implies that enhancement not dampening of CBET occurs. The preferred method to handle large intensities over significant interaction volumes is modeling pump-depletion and is currently being added to DRACO which doesn't require auxiliary saturation. Currently a modest form of saturation can be employed in DRACO to prevent excessive gain during convergence but typically is not required.

The resonant CBET gain region of the outbound rays in the backscatter mode never disappears but rather shifts into a smaller interaction volume because the relative instantaneous wavelength difference changes the ion-acoustic-wave–matching conditions of the interacting rays. The resonance region bifurcates and shifts both farther out in the corona (where the outbound rays have lower intensities and experience higher expanding fluid velocity and lower electron density) and closer inside the corona (where the interaction becomes shielded by the refractive shadow-boundary surface and/or outbound rays that have negligible intensity)¹⁴ [see Fig. 1(a)]. The net effect of wavelength detuning shifts the significant CBET-interaction region to lower Mach numbers covering both sub- and supersonic regions (e.g., 0.7 ≲ Mach ≲ 1.3.), i.e., moves the region closer toward the ablation surface. A sufficiently large wavelength separation (detuning) significantly reduces CBET exchange for direct drive by decreasing the interaction volume (e.g., ±6 Å UV). In contrast, an insufficient wavelength i.e., <2 Å UV, separation can lead to deposition and shell distortion via the forward-scatter mode.¹⁴ The efficacy of wavelength-detuning CBET mitigation diminishes as the plasma expands and the target implodes, which causes the CBET resonance regions to drift gradually into larger interaction volumes during the drive pulse.¹⁴ Selecting large wavelength-detuning values delay the inevitable onset of diminished mitigation and, for this reason, using the largest wavelength separation available yields the most favorable results.¹⁴ The decay rate of detuning efficacy differs according to the relative wavelength shift and causes asymmetries which are remedied using balanced designs. Simulations predict that wavelength-detuning CBET mitigation is effective for both symmetric direct drive (SDD) on OMEGA and polar direct drive (PDD), the present illumination configuration on the NIF, since the same mechanisms occur in both configurations, although the positive impact is more pronounced for PDD.¹⁴ Note that because the Randall-based CBET-model assumes a steady-state response, only the effects of wavelength separation can be properly treated and the dampening due to the coherence-disruption effects of bandwidth on LPI-process growth cannot be modeled. This also applies to the future modeling of smoothing by spectral dispersion (SSD);²⁶ see  Appendix B for a brief discussion.

The efficacy of wavelength-detuning CBET mitigation has been studied on the NIF.³¹ The target designed for these wavelength-detuning shots on the NIF was adapted from existing 600-kJ designs,⁹ where the trajectories and the shape of the imploding shell and scattered light were well described by the CBET model in DRACO. The platform target design is shown in the inset of Fig. 9, where the laser beam powers (shown in red) produce a peak overlapped intensity of ∼4–8 × 10¹⁴ W/cm² at the initial target radius.

The indirect-drive NIF beam geometry distributes 192 beam ports [grouped into 48 quads, shown as projected circles in Fig. 10(a)] toward the poles of the NIF target chamber, forming cones of quads that share a common polar angle.³² Repointing higher-intensity beams from lower latitudes toward the equator partially compensates for the NIF port geometry and higher incident angles when illuminating direct-drive targets. In this modified configuration, referred to as PDD,^24,33 CBET predictably dominates in the equatorial region where most of the crossing-beam interactions occur due to the requisite repointing,^14,34 as shown in Fig. 1(b). The beam port angles that would have induced significant polar CBET are absent in the PDD configuration. As a result, PDD implosions tend to become oblate because CBET reduces the laser drive preferentially in the equatorial region. With this motivation, a basic wavelength-detuning strategy exploits the PDD configuration, where each hemisphere has a different wavelength or color. However, the nominal symmetric wavelength mapping [see Fig. 10(a)] developed for indirect-drive targets precludes achieving true hemispheric wavelength detuning using typical PDD repointing configurations.²⁴ 

The NIF fiber front end³² supports three separate initial colors or wavelength shifts Δλ₀ = {λ₁,λ₂,λ₃} detuned from a central wavelength λ₀ ∼ 351 nm. Currently, the three-color {λ₁,λ₂,λ₃} mapping onto the NIF indirect-drive ports is symmetric about the equator [see Fig. 10(a)]. The symmetric color mapping precludes achieving a wavelength difference about the equator using typical PDD repointing schemes. A dramatic repointing (referred to as “cone swapping”) induces a wavelength difference within the equatorial region and when applied in either the northern or southern hemisphere [see Fig. 1(b) for the southern case]. For cone swapping, in one hemisphere the higher-latitude ports (“inner cones”: {λ₁, λ₂}) are repointed to the equator and the lower-latitude ports (“outer cones”: {λ₃}) are repointed to the mid- and high latitudes. For the first set of wavelength-detuning experiments described here, two different colors were specified such that λ₁ = λ₂ ≠ λ₃. The current NIF configuration, while not optimal, is capable of achieving a modest wavelength-detuning level Δλ₀ = {+2.3,+2.3,–2.3} Å UV, which is adequate for these proof-of-principle experiments. Cone swapping plus wavelength detuning induces the desired partial hemispheric wavelength difference between beams crossing the equatorial region. However, cone swapping is not intended for high-performance implosions because it induces a north/south asymmetry because of the combination of large repointing in only one hemisphere and the consequential illumination by nonideal spot shapes in the swapped locations (since DPP reconfiguration takes considerable operations time).

The far-field spot envelope (induced from DPP's²¹ and small-divergence smoothing) quad mapping is given by the current indirect-drive configuration on the NIF: the inner and outer cones. The inner cones (λ₁, λ₂; red/green projected circles in Fig. 10) use a wide elliptical spot shape not well suited for the equatorial region but appropriate for the polar and mid-latitude illumination; the wide spot has a super-Gaussian ∼3 profile with ∼2-mm-wide major axis and ∼4:3 ellipticity. The outer cones (λ₃; blue projected circles in Fig. 10) use a narrow elliptical spot shape adequate for the equatorial region but not well suited for polar or mid-latitude illumination; the narrow spot has a super-Gaussian ∼3 profile with ∼1.5-mm-wide major axis and ∼2:1 ellipticity. The values of the beam energy and repointing were additionally adjusted in the cone-swapping hemisphere to compensate for the swapped spot shapes and the higher incident angles using established PDD design principles.²⁴ The cone-swapping repointing scheme and the fixed DPP quad-mapping result in nonoptimal implosion symmetry [see Fig. 10(b)] because the illumination pattern in the cone-swapping hemisphere is nonideal and the incident angles are increased (decreasing hydroefficiency). As previously mentioned, however, cone swapping does provide the desirable equatorial wavelength separation of opposing beams from each hemisphere. For this reason, fusion yield and areal density are not metrics for these experiments, which concentrate instead on observables directly related to laser-energy absorption: implosion trajectory, shell morphology, and scattered light. Future reconfigurations (optimal DPP's for PDD,^21,24 flexible color mapping,^35,36 and larger wavelength separation^35,36) can relieve these constraints, and simulations predict improved overall fusion performance.

With this motivation, for the first time in direct-drive ICF, wavelength-detuning CBET mitigation was demonstrated and shown to improve energy coupling. The NIF PDD wavelength-detuning CBET mitigation campaign shots were performed in three pairs. Each pair consisted of one implosion backlit with ∼6.7-keV x rays produced from a planar Fe foil target energized by two quads of NIF beams with 45 kJ (see the blue curve in Fig. 9) of UV laser energy per beam with an equatorial view of the compressing shell. The second implosion of the pair examined self-emission images $(800 eV≤hν≤10 keV)$ of the compressing target from equatorial and polar views using a 25-μm Be filter. Additional diagnostics measured both hard x rays produced by energetic electrons arising from the stimulated Raman scattering (SRS) and possibly two-plasmon–decay (TPD) instabilities. The inferred levels of these instabilities contain at most only a few percent of the incident energy and do not affect the analysis of the laser–target coupling and CBET.⁸ The mitigation of CBET is expected to increase the observed levels of SRS and TPD because the resulting increased incident intensity could rise above the various LPI instability activation thresholds. Discussion of the observed increased TPD and SRS signals is planned for a future publication. It should also be noted that a larger wavelength separation than used in these experiments is expected to reduce TPD efficiency which implies that the TPD response should dampen rather than increase as inferred in these experiments given the larger bandwidth.⁴⁷ 

The first pair of control experiments (shots N160405 and N160406) with the same wavelength for all the beams (zero detuning) was performed to establish the baseline experimental observables. Next, two pairs of experiments with a detuning mapping of Δλ₀ = {+2.3,+2.3,–2.3} Å UV were performed to evaluate the efficacy of wavelength-detuning CBET mitigation. The zero-detuning and first-detuning shots (N160821–001 and N160821–002) employed southern-hemisphere cone swapping, as illustrated in Fig. 10(b). The second-detuning shots (N170102 and N170103) applied northern-hemisphere cone swapping, primarily to effectively image the self-emission from the opposite/antipodal pole and to observe the expected image inversion. The repointing (accounting for mirror-image cone swapping) and pulse shapes were nominally identical for all shots where the only intended difference was the wavelength configuration.

The simulated and measured backlit gated x-ray radiographs are analyzed to show shell morphology evolution and in-flight shell trajectory, which are both used to infer energy coupling. The gated images (gate time ∼100 ps) shown in Fig. 11 compare the shell morphology for the three backlit shots. The experimental framing-camera images are a composite of several images close in time for this slowly moving target that were cross-correlated and adjusted for magnification to enhance the signal-to-noise ratio; the measurements used a 30–μm pinhole. The simulated images are post-processed with the x-ray imaging code Spect3D³⁷ with pinhole and gate values matching the experimental setup. The first two rows are radiographs of matched post-shot simulations and experimental results for the baseline zero-detuning and wavelength-detuning shots with southern-hemisphere cone swapping. The last row shows radiographs for detuning shots with northern-hemisphere cone swapping. All the backlit radiograph data show remarkable agreement between simulation and experiment, especially the expected trend for the detuning shots. A mere ∼2% to 3% additional laser energy is absorbed with detuning, but since this energy is localized to the equatorial coronal volume fraction (∼25%), and the deposition is redistributed closer to the ablation surface increasing hydrodynamic efficiency, the result is dramatic as observed with the gated x-ray radiographs.

Most notable was the design prediction and the subsequent measurement of the equatorial mass accumulation with active wavelength detuning (bottom two rows in Fig. 11). As predicted, the mass accumulation flipped orientation when cone swapping was applied to the opposite hemisphere. The wavelength-detuning design attempted to minimize the ℓ = 2 Legendre mode while accounting for the spot shapes, pointing, and energies in conjunction with the expected increased drive in the equatorial region caused by CBET mitigation. The equatorial mass accumulation is a common feature in PDD implosions with small ℓ = 2 contributions and not directly related to CBET mitigation. The accumulation is caused by unbalanced lateral mass flow toward the equator (from primarily oblique incidence) when sufficient equatorial drive is available (e.g., from CBET mitigation) while achieving a small ℓ = 2 component. Corrections for the equatorial mass accumulation consist of design attributes that specifically address the equatorial region, e.g., using optimal DPP's or shell-mass contouring.

The shell trajectory is inferred from the simulated and experimental backlit radiographs by first extracting the outer steepest gradient surface or radii [see Figs. 11 and 12 (inset)]. The majority of the CBET gain occurs in the equatorial region [Fig. 1(b)] and consequently the region expected to benefit from wavelength detuning. Both the surface-area–weighted average of the entire extracted surface and a range restricted to the equatorial region (shown here) demonstrate the benefit. When the extracted shell surface is restricted to the equatorial region (±30° region about the equator) and plotted as a function of time (see Fig. 12), the inferred implosion speed increases as a result of wavelength-detuning CBET mitigation. The equatorial shell speed increases 9% from 144 to 157 μm/ns based on simulation (experimentally a 16% increase from 133 to 154 μm/ns) because wavelength-detuning CBET mitigation deposits ∼3% additional energy within the small volume over the equator with enhanced hydroefficiency due to coronal deposition closer to the ablation surface. The enhanced equatorial velocity consistently results when comparing the extracted outer shell contours taken from zero detuning and detuning shots in Fig. 12 (inset), where the entire surface-area–weighted average implosion speed increases experimentally by 13%. This observation in simulation and experiment suggests that CBET mitigation via wavelength detuning promises to alleviate the excessive losses in directly driven ICF targets. Simulations predict that as the wavelength separation increases so does the efficacy of the mitigation.³⁵ Simulations also predict that large wavelength separations (±6 Å UV) also significantly benefit implosions with ignition-relevant implosion speeds (∼350 to 400 μm/ns) (Ref. 38).

The simulated self-emission images in both the polar and azimuthal views predict the presence of a core self-emission feature; see Fig. 7. The corresponding experimental self-emission measurements confirm the presence of this self-emitting core feature. In fact, only this series of PDD shots using wavelength detuning consistently measure this long-duration (∼400ps) feature on every self-emission measurement shot. The self-emitting core signature for wavelength detuning shots help qualify current 2-D and future 3-D simulations as well as provides an excepted diagnostic marker of the wavelength detuning shots.

Additional NIF capabilities improve PDD target–energy coupling according to simulations. The proposed capabilities include flexible color-to-port mapping, custom DPP's, and an increase in wavelength separation. Flexible color-to-port mapping eliminates cone swapping and consequently improves the north/south asymmetry. Providing a full set of optimized SMA-DPP's mounted to the ideal port locations minimizes repointing (assuming flexible color remapping) and further improves the north/south symmetry. SMA-DPP's also improve energy efficiency (by eliminating over-the-horizon energy dumping caused by repointing), improve uniformity, and increase CBET mitigation (by tailoring the SMA to be slightly interior to the initial target radius) (see Fig. 13). Increased wavelength separation provides improved CBET mitigation and the flexible color remapping permits a balanced tri-color configuration (see Fig. 14).^35,36 Simulations of intermediate-energy (∼700-kJ) designs on the NIF illustrate the expected improvements over these proof-of-principle experiments with additional NIF capabilities (see Fig. 15). The proposed additional NIF capabilities provide dramatic symmetry control and higher convergence and demonstrate control over the equatorial mass accumulation. The progress toward providing these additional NIF capabilities can be taken in a staged approach where, for example, one-third of the total SMA-DPP's are used to populate the beam ports repointed to the equatorial region as an initial step toward the ultimate goal.

The OMEGA Laser System employs a three-way equal-energy split initiating the eventual separation into 60 large-area beams. Each leg of the three-way split contains 20 beams distributed around the OMEGA target chamber in a manner that minimizes the nonuniformity for each leg separately (see Fig. 16). Replacing OMEGA's large-aperture ring amplifier (LARA) and three-way split with three independent vertical amplifier sections (similar to the NIF) would outfit OMEGA with three independently controllable wavelengths in sets of 20 beams [originally proposed internally at the Laboratory for Laser Energetics (LLE) circa 2013]. Phase modulation generates multiple coexisting spectral lines without the severe coherent interference expected by co-propagating beams with different wavelengths. Phase modulating each beam to create widely separated lines would be superior to three sets of 20 beams since it effectively multiplies the total number of interacting sources and lowers each source line's intensity, which also lowers CBET interaction strength in concert with wavelength separation. However, this option requires substantial research to achieve the required line separation and to manage any issues amplifying this bandwidth (e.g., large-scale optical parametric amplification), whereas the option proposed here is feasible with currently available optics. Note that depending on the wavelength separation and desired shot energy, this potential upgrade could require wide-bandwidth optical coatings, new amplifier glass, and frequency-conversion crystals. While the original intention of the spatial distribution of the legs was not to support multiple colors, the spatial distribution of the legs is advantageous for CBET wavelength-detuning mitigation in the symmetric direct-drive configuration because it creates a distribution of wavelengths around the target that reduces the CBET interactions that occur using dissimilar wavelengths.

A cryogenic DT target matched with a triple-picketed 21-kJ pulse provides a platform to test multiple CBET mitigation schemes via simulation (see Fig. 17), where the target's nominal initial outer radius is r_t = 415 μm, which includes the 80-μm DT ice layer and plastic (CH) overcoat. The energy was lowered for wavelength detuning, which was projected to be limited to 21 kJ with the current amplifier-gain window, although it is expected to have full system energy with an upgrade to wider-bandwidth laser glass.

The test platform uses a pulse and cryogenic DT target designed to produce high compression and hot-spot pressures exceeding 100 Gbar in reference simulations without including CBET. The hot-spot pressure is defined as the peak neutron-weighted total pressure attained interior to the compressed shell during the implosion. The value of 100 Gbar qualifies the implosion as ignition-scale–relevant conditions and is currently a major goal of the Omega Laser Facility. The non-CBET reference simulation uses a spot shape labeled SG5, which is defined as a super-Gaussian (SG) order equal to 5.0 with 95% enclosed energy set to r_t (equivalent to a 5% intensity point set to 466.9 μm). The non-CBET simulation becomes the reference and goal that effectively measures success of any CBET-mitigation scheme because any successful CBET-mitigation scheme should converge toward this non-CBET simulation, i.e., the maximum expected laser absorption given that DRACO only simulates IBS and CBET. Another important reference simulation enables the CBET effect without employing any mitigation scheme that measures the overall impact of CBET, i.e., the minimum expected laser absorption, and also employs the SG5 spot shape. The significant CBET-interaction region for this reference case tends to be supersonic (1.0 ≲ Mach ≲ 1.5) during the main pulse. These two simulations without and with CBET set the upper and lower laser-absorption bounds, where their difference sets the maximum recoverable energy that can be expected and this difference can be used to measure the success of a particular mitigation scheme.

The two CBET mitigation schemes compared in this paper are a wavelength-detuning scheme with customized SMA-DPP's and a spatial scheme that underfills the target by 25% by decreasing the diameter of a typical far-field spot shape which is typically denoted as R75. Note that both of these mitigation schemes employ the spatial domain mitigation scheme to varying degrees. The degree of spatial mitigation directly impacts target performance, which can undermine any resulting benefits of CBET mitigation.

The first CBET mitigation scheme employs wavelength detuning with a customized SMA-DPP. The wavelength-detuning configuration is distributed as shown in Fig. 16, and the wavelength-detuning configuration is $Δλ0={−6,0,+6}Å UV.$ The wavelength-detuning case also employs SMA-DPP's labeled SMA90SG3.3 to optimize the on-target uniformity while gently removing some over-the-horizon energy to spatially mitigate CBET. The SMA-DPP's were designed for a slightly overfilled SG = 3.3 with 95% enclosed energy set to r_t (equivalent to a 5% intensity point set to 457.4 μm), which tends to minimize on-target uniformity, and using another super-Gaussian modulation that apertures the spot tails to trim the energy at 0.90 * r_t while maintaining the underlying optimal super-Gaussian 3.3. The significant CBET-interaction region for this detuning case is shifted to lower sonic speeds and is approximately centered on the Mach-1 surface (0.7 ≲ Mach ≲ 1.3) during the main pulse.

The second CBET mitiagation scheme to be compared employs a spot shape that underfills the target by 25% (R75). The reduced spot shape specifically examined in this paper is denoted as SG4-R75 (Ref. 13) because the spot's SG = 4.0 with 95% enclosed energy set to 315 μm (∼0.75 r_t; equivalent to a 5% intensity point set to 352.0 μm). There are a few detrimental side effects on target performance that result from underfilling the target by too much. The SG4-R75 spot has a deleterious impact on target performance by increasing the target's sensitivity to imbalances (such as mistiming, mispointing, and offset) and increasing the imprint susceptibility (short-wavelength modes imprint on the ablation surface caused by the speckled far-field pattern of a DPP) because of the lower effective number of overlapped beams; also the CBET activity interior to the beam increases because the beam intensity increases for the same delivered energy. The SG4-R75 case is more sensitive to laser imbalances (typical OMEGA values: 10% rms constant power imbalance, 5-ps rms mistiming, and 10-μm rms mispointing), as illustrated in Fig. 18 for power imbalance, where the induced modal structure is larger for ℓ < 10 Legendre modes.

The results from all four runs are tabulated in Table I, where the first two rows are the reference runs that bracket the CBET effect. The non-CBET run absorbs 78% of the laser energy and achieves the desired goal of >100-Gbar hot-spot pressure, where the achieved pressure is 169 Gbar by design to account for laser system imbalances and because CBET mitigation schemes are not 100% effective. The CBET effect results in a decreased absorption fraction of 22.9% energy difference that degrades the hot-spot pressure ∼5×, which is $≪ 100 - Gbar$ goal. The 22.9% energy difference between CBET and non-CBET runs can be used to measure the success of a mitigation scheme using the same target design. The SG4-R75 DPP helps to mitigate CBET, increases the absorption fraction to 65.2%, and more than doubles the hot-spot pressure to 72 Gbar, which is still under the design goal of 100 Gbar using a simulation under ideal conditions. The CBET mitigation of SG4-R75 yields a (65.2–55.1)/22.9 = 44.1% mitigation success factor. The SMA90SG3.3 simulation further improves the CBET mitigation, bringing the absorption fraction to 73.6% (nearly equivalent to the non-CBET run) and the hot-spot pressure above the design goal at 119 Gbar. The CBET mitigation of SMA90SG3.3 yields a (73.6 – 55.1)/22.9 = 80.8% mitigation success factor.

The resultant shell morphology is illustrated by plotting the mass density near peak compression (see Fig. 19). Peak compression occurs at t ∼2.9 ns for the non-CBET reference run and the SMA90SG3.3 because the absorption fractions are similar, whereas it occurs at t ∼3.05 ns for the SG4-R75 run because of its lower energy absorption. Since the CBET mitigation for SMA90SG3.3 was nearly complete, the shell morphology is similar to the reference case with the exception that it is slightly delayed and the plotting resolution was not fine enough to capture it closer to its true peak; therefore, the lower mass density shown in Fig. 19(b). The ℓ = 10 Legendre mode is more pronounced in the SMA90SG3.3 case because of the use of SMA-DPP's with 90% sub-aperturing, which was expected but deemed acceptable for the benefit of improved CBET mitigation. However, the ℓ = 10 Legendre mode is far more severe for the SG4-R75 case and is the primary reason for its poor performance. The CBET mitigation benefit of the SG4-R75 spot does not outweigh the damage done to the shell. While it may be possible to fine-tune the pulse and target for higher hot-spot pressures using an R75 type of DPP just as SMA90SG3.3 can be retuned for the improved performance, this was not the point of this comparison. The point was to ascertain the mitigation success factor for different mitigation schemes by varying nothing except the mitigation scheme since success is determined relative to the non-CBET reference run. The fact remains that the SG4-R75 mitigation success factor is ∼0.5× that of SMA90SG3.3. This testing platform technique can be used to evaluate a variety of CBET mitigation schemes and is not limited to the two cases performed here.

To estimate the impact of laser system imbalances on each CBET mitigation scheme, more simulations were run that include mistiming = 5-ps rms, mispointing = 10-μm rms, and a constant power imbalance = 10% relative to the nominal values, where random numbers are drawn from a normal (Gaussian) distribution. The random numbers drawn for each simulation are repeatably random; therefore the same set is used for the sake of a fair comparison. The resultant shell morphology is illustrated by plotting the mass density near peak compression (see Fig. 20). The results from these runs are tabulated in Table I; the values of which are within the brackets in the third and fourth rows. The SG4-R75 case has lower hot-spot pressure and yield, while the SMA90SG3.3 still achieves the >100-Gbar goal despite being lowered by the imposed laser imbalances.

Draco has the ability to emulate the effect of the non-zero m-modes by projecting the effects of the various laser imbalances onto a hardsphere while next performing a spherical harmonic decomposition on-the-fly during the simulation, extracting the power of the non-zero m-modes and then subsequently imposing this power via an m = 0,ℓ-mode laser perturbation corresponding to every ℓ-mode supported by the mesh resolution. This is typically considered an over-estimation of the true non-zero m-mode effect on target performance because all the power is presented as a single mode instead of spread over many m-modes. The effect of non-zero m-modes on the target performance using this method is tabulated in Table I although not illustrated in the plots. An inspection of the tabular data suggests, at least as far as modeled in a 2-D code, that the detrimental effects on target performance due to laser imbalances are of the same level as the degradation due to the SG4-R75 spot-shape. For example, the neutron yield degrades by about a factor of 3× going from SMA90SG3.3 to SG4-R75 for each of the three simulations examples and the degradation is also ∼3× when comparing either design and applying laser imbalance with non-zero m-modes. The effects of the various laser imbalances may be larger due to the smaller surface area of true non-zero m-modes and their potential faster growth if the power was properly accounted for running a 3-D simulation; this comparison is planned for future research. The dominant detrimental effect of the small spot shape alone was sufficient to demonstrate the clear advantages of using SMA spot-shapes and wavelength detuning while attempting to account for laser imbalances. This is not to argue that laser imbalances are unimportant but rather to illustrate the effects of the smaller spot shape relative to laser imbalances. The lower performance using the non-zero m-modes suggests that an improved target and pulse shape design would be prudent, e.g., using a large pulse energy.

In conclusion, the first direct-drive wavelength-detuning CBET mitigation experiments on the NIF with a modest wavelength difference between crossing beams confirmed improved coupling as predicted by multidimensional hydrodynamic simulations. The improved coupling was inferred from the faster shell trajectory. A increased array of scattered-light detectors could be used on the NIF to measure improved absorption since the current number of detectors are too few for an accurate inference considering the amount of azimuthal variations. These direct-drive proof-of-principle experiments are the first such experiments and provide a path forward to recovering the energy loss caused by CBET. Simulations predict that as the wavelength separation increases (e.g., the ±6-Å UV predicted NIF full-energy limit¹⁴), the equatorial drive continues to improve and requires rebalancing to minimize ℓ = 2. Simulations also indicate that judicious use of all three colors with flexible color mapping in the fiber front end on the NIF produces better-balanced CBET mitigation designs in PDD.^14,35,36 Simulations predict that symmetric direct drive on OMEGA will benefit from wavelength detuning since its three main driver legs already distribute adequately over the target. Additional CBET mitigation domains may be combined with wavelength detuning, e.g., optimized spot shapes that reduce the laser energy refracting over the horizon while maintaining optimal shape (SMA).²¹ Future experiments are planned to scope out the capabilities of wavelength-detuning CBET mitigation to further improve coupling and to address the asymmetry by proposing system changes to both OMEGA and the NIF: adding multiple wavelength sources to OMEGA, expanding the NIF's wavelength-detuning range, using SMA-DPP's, different wavelengths within NIF's quads, and remapping the NIF fiber front end to obviate cone swapping.

This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0001944, the University of Rochester, and the New York State Energy Research and Development Authority.

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

The Mazinisin 3-D ray trace is part of the multiphysics radiation–hydrodynamics ALE 2-D code DRACO. Each physics package solves its own differential equations by employing the common split-step technique, where other physics effects are considered frozen or constant, while the current physics package is active during the discrete time step of the simulation. Typically, the result of one physics module builds a source term that drives a subsequent module, e.g., the ray trace stores the absorbed laser energy, which is then used to drive the thermal motion of the electrons and ions. Once all physics modules are completed, the time step is advanced and the process is repeated. The ray-trace package is highly scalable to run in parallel (with tested scalability to ∼1200 cores without any noticeable saturation) via the domain decomposition package Daashkaa,³⁹ which is integrated into Mazinisin; this combination affords the CBET package Adaawam the necessary scalability and reduced memory footprint.

The Mazinisin package propagates refracting rays in 3-D within a 2-D plasma atmosphere and accounts for arbitrary multiple beam orientations, arbitrary far-field spot-shapes, independent pulse shapes, beam repointing, target offset, laser absorption, frequency chirp, speckle imprint, and smoothing by spectral dispersion (SSD)²⁶ and for many nonuniform and random beam-to-beam variations such as pulse power imbalances, mistiming, or mispointing, as well as the 3-D CBET interaction effects (using Adaawam) with or without speckled beams. DRACO is able to emulate the effect of 3D modal power induced by laser imbalances by imposing any non-zero m-mode spectral power into a corresponding m = 0,ℓ-mode perturbation. Since the radiation-hydrodynamic code DRACO is only 2-D, the 3-D spatial feedback into the corona cannot be modeled; however, some useful 3-D scattering and deposition effects of refraction and CBET are captured within diagnostic output and can be correlated to experiment.

DRACO simulates ICF implosions of spherical targets while being a 2-D code. The logical mesh for a spherical object should reflect the symmetry of the object. The type of mesh that DRACO employs is an r–z polar cylindrical mesh divided into quadrilateral computational cells or zones, where the orthogonal cylindrical coordinate system maps as $R≡x2+y2$ and Z = z, where ${x,y,z}∈ℝ3$ define the 3-D Cartesian coordinates. The polar mesh in the R–Z plane is considered rotated 2π azimuthally, $Φ$⁠, about the symmetry axis $ẑ$ to form the complete initially spherical object, where the 3-D cylindrical coordinates are ${R,Z,Φ}.$ Both the Cartesian and cylindrical 3-D coordinate systems are related to the orthogonal spherical coordinate system ${r,θ,ϕ},$ where angles θ and ϕ are referred to as the polar and azimuthal angles, respectively. The equivalences or transformations between the three 3-D coordinate systems are given by

and where the first two equations in Eq. (A2) imply

where ϕ = tan⁻¹(y/x) and θ = sin⁻¹(R/r), but the ratios given in Eq. (A3) are more efficient if angles are not required.

The Lagrangian aspect of DRACO means that the underlying mesh flows with the fluid which, in general, alters the mesh from an orthogonal grid. This implies that ray tracing must be performed within a nonuniform grid precluding any simplifications or high-efficiency methods provided by regular orthogonal meshes. Regardless, a successful ray trace must be efficient and low noise because the laser is the energy source that ultimately drives all phenomenon in an ICF implosion, which is inherently an unstable hydrodynamic problem that amplifies perturbations whether they are intentional or not.

A ray trace can be divided into three broad classes of problems to solve that are equally and deceptively complex because of the myriad of issues that can occur on an irregular mesh; (1) ray initialization and entrance-cell ray entry, (2) refractive ray propagation with IBS and CBET, and (3) cell exit and cell-to-cell transfer. Each problem class contributes appreciably to defining a high-quality algorithm that generates low noise and high accuracy deposition. The second problem class differentiates itself as being the most time consuming but it is not the most important. Ignoring the importance of each class will lead to unsatisfactory results like noise-dominated shell morphology.

The first problem class involves selecting the initial ray position, direction, and energy as well as determining the cell-entry position. The choices made here significantly affect the noise induced in the simulation. Judiciously selecting the ray positions, for example, overwhelmingly impacts subsequent binning noise in the simulation. Before any results are discussed, the other two problem classes must be addressed because the results of the first problem class are inherently coupled to the next two problem classes.

The second problem class involves 3-D refractive ray tracing with CBET and IBS effects through an arbitrary quadrilateral mesh; currently, these are the only two LPI processes simulated in DRACO. The CBET effect does not alter the ray trajectory directly within a time step and does not come into play until after the ray traverses a computational cell because of its high computational costs. After the ray trajectory is formed the CBET coefficient and frequency chirp integrated along the path are determined, CBET is calculated as described in the second section [Eqs. (2)–(5)] using either a single or multiple path segments depending on the curvature. The calculation of IBS will be discussed alongside ray tracing. The discussion of frequency chirp is in  Appendix B. Mazinisin propagates rays in 3-D through the azimuthally symmetric 2-D atmosphere. Since the underlying physical quantities that govern the ray trace are defined on a 2-D R–Z polar cylindrical mesh and the rays propagate in 3-D Cartesian coordinates a mapping to and from the cylindrical coordinates, is required to interpolate the physical quantities as a function of any ray's position vector r.

At the heart of any ray trace is the nonlinear partial-differential equation called the eikonal equation $|∇S|2=nrefr2$ (Ref. 40), from which one can derive the ray equation, viz.,

where r is the position vector and the differentials are taken with respect to the elemental propagation path length ds. The ray equation may be rewritten as

where the two vectors are identified as the unit tangential vector

and the curvature vector of the ray $K≡d2r/ds2,$ while the directional derivative of the refractive index is given by $dnrefr/ds≡t̂⋅∇nrefr$ (Ref. 41). The unit tangential vector $t̂$ is also known as the propagation vector since it defines the propagation direction. The curvature vector is related to the propagation vector as

and consequently obeys $K⋅t̂=0$⁠. Solving Eq. (A6) for the curvature vector yields the exact expression

where $nrefr≡1−ne′$ (Ref. 42), $∇nrefr,$ and $∇ne′$ are all functions of the position vector r. In general, the solution to Eq. (A6) is analytically intractable, but numerical solutions are possible.

For example, through linearization of the position vector and tangential vector via a truncated Taylor series expansions about the current position and propagation vectors r₀ and $t̂0$ over a small step size ds, viz.,

where the unit tangential vector $t̂$ defines the propagation direction. A first-order (in error) Euler quadrature algorithm (Refs. 41 and 43) is defined by evaluating Eq. (A10) at the current position and propagation vectors r₀ and $t̂0$ to obtain an updated curvature vector K, which are then substituted into Eqs. (A11) and (A12) for a small step size ds to obtain the next position and propagation vectors. The error is controlled by setting the step size ds using $|K|,$ viz.,

where ϑ_comp is a user-defined compression factor that controls the accuracy, L_char defines the characteristic size of the computational cell, and ε is the machine precision. The first-order method is adequate for regions of low curvature, i.e., low $|K|$ out in the corona $ne′≤0.2,$ because to get reasonable accuracy in high-curvature regions, the compression parameter ϑ_comp must be set so small that it is computationally more efficient to use adaptive integration schemes. In addition, the first-order method can have difficulty turning near critical, i.e., regions of very high curvature, which leads to deposition noise and rays hitting the critical surface even when using small ϑ_comp values.

Higher-order (in error) adaptive quadrature methods resolve many issues relating to accuracy, efficiency, and high curvature. In addition, it becomes straightforward to include adaptive IBS integration, further increasing accuracy and efficiency consistently with ray trajectory integration. Any ordinary differential equation (ODE) can be reduced to a set of coupled first-order differential equations and written generically as $dyn/ds=fn(s,y1,…,yN),$ $∀n∈{1..N},$ where the functions $fn(s,y1,…,yN)$ define the known derivatives to be evaluated along the integration path. The coupled set of first-order differential equations defined for the ray trace are given by the seven-element variable vector

and the corresponding seven-element derivatives vector

where $ξ$ is the optional IBS absorption kernel and is defined as^42,44

and arranged with all constants toward the left and spatial variables toward the right, where k_B is the Boltzmann constant and $ln Λe$ is the spatially dependent electron Coulomb logarithm. Equations (A14)–(A16) form the basis of high-order adaptive quadrature methods.

In an adaptive quadrature method, an ODE driver routine given in Ref. 43 interfaces with an adaptive ODE stepper routine to solve Eq. (A14) by evaluating the derivatives from Eq. (A15) from a fourth- or fifth-order algorithm (such as the Cash–Karp, Runge–Kutta method⁴³). The adaptive stepper adjusts the steps over an integration interval $[s1,s2]={sn∈ℝ;s1≤sn≤s2},$ where $sn≡sn−1+ds$ defines a variably spaced sequence taking large ds steps over low gradient terrain and iteratively refined small ds steps as the gradients increase in order to maintain a specified level of accuracy. The result is an efficient algorithm that takes the smallest number of steps and, consequently, function evaluations while maintaining a specified accuracy. This is especially important in high curvature regions where the ray trajectory is making the tightest turns and in this region depositing the highest energy fractions. Low-order methods, e.g., parabolic trajectories or the Euler quadrature, cannot prevent hitting the critical surface or making deposition errors because the gradients are not constant over a computational cell, and in high-gradient regions, a low-order method requires exceedingly small step sizes, making it impractical. However, using adaptive quadrature succeeds in this regard because this method adapts to spatially varying gradients, which leads to more-accurate trajectories and deposition while ensuring that rays never hit the critical surface (except for rare degenerate normal incidence cases, which occur only for one ray per beam in a radial plasma); see Fig. 21.

The ODE driver routine requires modification to handle situations found in ICF implosions and the distorted meshes resulting in ALE hydrocodes. The driver must be able to call different types of stepper routines such as the Runge–Kutta or Bulirsch–Stoer methods⁴³ to take advantage of their different strengths. (DRACO offers both user control as well as an automatic selection capability.) The routine also needs flexibility to be called for many types of integration, i.e., ray trajectory, absorption, or any other ODE equations appropriate to the ray trace. The driver also must be able to temporarily guide a ray past critical while it converges to the correct trajectory that avoids critical. Going past the critical density is allowed by using appropriate asymptotic limits that avoid triggering a stiff-equation response, i.e., using the machine precision ε in refractive index formula, which smoothly goes to this limit. Another adaptation allows the algorithm to dynamically decide the most-efficient integration order to apply, which boosts ray-tracing efficiency while not sacrificing accuracy. The most-important adaptation is limiting the overall path length when crossing the cell-edge boundary or up to any singularities such as the critical surface.

Numerical integration of Eq. (A6) requires repeated interpolation of the electron density and its gradient along the ray trajectory when the underlying functional form is not known. Likewise integrating Eq. (A15) requires continuous interpolation of the kernel Eq. (A16) and not its individual components because this leads to scalloping when different interpolated functions are multiplied. Cell-centered quantities are first interpolated onto the cell vertices using volumetric mass-weighted averaging. The plasmas being modeled are naturally continuous and likewise the interpolator should mimic this continuity and not lead to unphysical jump conditions across cell edges. Isoparametric mapping within the quadrilateral cells satisfies this desirable and essential condition and is employed in Mazinisin. The basic isoparametric mapping can be extended to include radial dependence other than linear, e.g., an exponential or 1/r² radial decay of the electron density can be employed to provide larger cells with an improved accuracy that reflects the assumed radial functional form.

Higher-order (referring to the expansions and not the quadrature type) methods can be used to solve Eq. (A6), but linear methods tend to be superior because they involve many simple calculations that can achieve similar accuracy but in less time because of the computational overhead of the higher-order methods. The seemingly mathematical elegance of higher-order methods is outweighed by the computational complexity. Higher-order trajectories intersecting with nonlinear boundaries imply high-order roots, which increases solution complexity and computational overhead for no real gain when lower-order [i.e., Eq. (A15)] suffice and yield high accuracy and high efficiency. In addition, using a paucity of rays in lieu of high density of rays may seem like an attractive option at first, however, it suffers by trading off efficiency of propagating of many simple ray trajectories with the consequential inaccurate volume interpolation and complicated intersection calculations for deposition especially where it matters the most in ICF implosions, i.e., near critical.

The third problem class involves cell exit and cell-to-cell transfer. Cell-edge crossing detection uses a metric or “ruler” to measure the inside/outside condition at a cell's edge; the same metric is applied from any cell and during propagation and this consistency rules out machine or metric precision errors. Since neighboring cells share the same cell edge, the same detection response for a given cell face is guaranteed regardless of the cell sharing the cell face. Even if the metric has slight numerical errors, e.g., because of low accuracy or machine precision, it will not matter with regard to cell-edge crossing detection since the answer is absolute and invariant of which cell queries the inside/outside condition. This holds true because each pair of neighboring cells share the same interface and its accompanying functional form and therefore the same answer is absolute and guaranteed regardless of any small errors inherent in the cell-edge detection metric. The only possible side effect of numerical errors in the cell-edge detection is slight deposition errors but this is controllable and vanishingly small to machine precision by appropriately setting the detection tolerance. Cell-edge detection uses a root-polishing/root-finding algorithm (Brent's method⁴³) that has been modified to bias the resulting position to be just outside the current cell; this technique is powerful and efficient. Use of bracketed roots guarantees the solution of finding a ray's exit; the side benefit is that rays are never lost or fail to propagate. There does exist, however, the possible trivial root where the chosen ds [Eq. (A13)] lands exactly on the cell's edge, which can be biased to prevent this situation at an additional computational cost. The trivial root eventually happens (albeit rarely) when propagating trillions of rays over thousands of time steps in thousands of simulations; the rare trivial root is solved by nudging the ray by a small step to force a positive cell-edge crossing condition. The adaptive root finder allows the hyper refinement of the step size to bring the ray arbitrary close to the cell's edge while ensuring cell crossing; the side effect is reduced energy binning noise. Another anomalous situation that can occur for unphysical meshes causes a ray to be infinitely guided internally between two density peaks without losing energy; unphysical mesh detection is necessary to prevent this unphysical situation. Cell-to-cell transfer is somewhat trivial since the mesh topology is known and each neighbor is known across the cell edge. The possible transfer across a cell corner is robustly handled automatically without the need for complicated corner detection logic. For instance, if the transfer into the edge-based neighbor is done and the corner crossing has occurred, this next cell would immediately trigger a cell crossing, which would immediately transfer the ray into its cell-edge neighbor, effectively transferring the ray across a cell corner. Removing unnecessary and complicated logic like the corner-crossing detection makes Mazinisin's algorithm robust and efficient. This also implies that rays are never lost since cell exit is guaranteed.

Now that the three problem classes have been outlined, the results of the adaptive inverse projection algorithm can be discussed. The adaptive inverse projection algorithm is used to initialize ray starting points, which encourage maximal deposition into the intended cells, greatly reducing binning noise see (Fig. 22). The method adapts to shell distortions as well as refraction to minimize binning noise. Compared to a random distribution with the same number of rays chosen randomly each ray-trace step, the different approaches have very different noise backgrounds. The random distribution technique relies on the incoherent addition of the noise patterns from different time steps that ideally dampens the noise response below the applied signal. In practice, however, to achieve the same level of noise reduction, a random method would have to increase the ray density many fold per time step (or equivalently increasing the number of ray-trace steps) because the insufficient smoothing obtained by averaging random distributions has diminishing returns. The use of an inverse projection algorithm achieves low noise through the judicious choice of launch positions that tend to deposit the bulk of energy in the intended cells, thereby minimizing binning noise.

The basic inverse projection algorithm determines the ray launch positions for each beam port starting at cell positions specified within the 3-D plasma mesh, as position vectors $rp∈ℝ3$⁠. The beam port is defined as a position/direction vector $P=RpP̂,$ where $Rp$ is a radius just outside the maximum extent of the plasma and the beam-port unit vector

which points toward the beam port's central axis and is defined with the beam port's spherical coordinate angles ${θPrt,ϕPrt}.$ A straight-line back projection along the trajectory given by the beam-port direction vector maps these points into the far-field plane of each beam port [see Fig. 23(a)], viz.,

where $rff∈ℝ3$ under the condition that $0<N<R p,$ where the range restrictions N > 0 define the visibility of r_p from the beam port P and $N<R p$ results from the far-field plane's definition. A far-field transformation maps the position vector r_ff into the local 2-D Cartesian coordinates of the beam port's far-field plane, viz.,

where the far-field transformation matrix is given by

and $rFF∈ℝ2$⁠, by construction, since the transformation maps into a plane. The mapped far-field coordinate determines the potential ray's intensity by interpolating this position into the beam's intensity pattern. The launch positions in the far-field plane form concentric ellipses [see Fig. 23(b)]. Each starting cell position in the plasma has an associated elemental surface area tiled together with all the neighboring points. The dot product of the beam-port unit vector and the surface area outward normal determines the points' visibility (when positive definite), the projected area onto the far-field plane, and thereby the power to be given to the ray by multiplying the visible ray's area and intensity. The starting positions in the 3-D plasma form a closed and connected surface subtending 4π steradian that projects a continuous and connected surface area onto the far-field plane; e.g., if the surface is spherical with radius r_s, this maps onto a $πrs2$ area in the far-field plane. The first inverse-projection surface is typically chosen near critical since this region requires the lowest noise because it gives rise to the highest hydroefficiency, i.e., noise near critical adversely impacts target imprint more than noise far out in the corona. Once the plasma has expanded, the first surface provides insufficient coverage of the plasma back-projected onto the far-field plane. Other inverse projection surfaces are then selected to provide complete coverage of the full plasma extent. When the next subsequent surface area A_next is back-projected into the far-field plane, only the relative complement set $(Anext/Aprev,$ where A_prev is the previous back-projected area) is allowed to propagate.

As the plasma expands, the importance of refraction begins to play a role. Refraction causes the ray intended to deposit in one cell to deposit in a neighboring cell, leading to binning noise. An algorithm that adapts to the increased refraction effect becomes necessary to reduce noise. If the projected ray intensity (onto the inverse projection surface) is accumulated and plotted by impact parameter (dot product of the ray direction and a sphere's outward normal), the distribution peaks for a particular impact parameter. These rays become the most-significant players that determine the extent of binning noise. Minimizing the binning noise for these rays will minimize the noise for the total deposition. With this in mind, an adaptive algorithm optimizes a search for a surrogate inverse-projection surface that will minimize binning noise. The optimal surface is found when these rays are back-projected from this surrogate inverse projection surface and are then turned in the intended cell when propagated with refraction (see Fig. 24). Since the greatest energy is deposited where a ray turns, binning noise is minimized for the rays with the impact parameter corresponding to the peak of the distribution. The remaining rays surrounding the peak also have their binning noise reduced when using the surrogate inverse projection surface. The surrogate surface lies interior to the intended inverse projection surface. This adaptive algorithm works well for 2-D hydrocodes and optimizes the surrogate surfaces as the plasma evolves. The results of the adaptive inverse projection algorithm are compared to the basic algorithm (see Fig. 25). The binning noise is noticeable in Fig. 25(a) early in time, cf. to Fig. 25(b), and this excessive noise imprints in the shell and develops mass perturbations that persist late in time. These mass-density perturbations are remedied using the adaptive algorithm. Improved algorithms are being developed that account for refractive back-propagation that will further reduce binning noise and apply to 3-D hydrocodes.

The severity of residual noise induced from a random ray distribution versus the adaptive inverse projection algorithm is demonstrated using imprint simulations. Imprint is modeled in a NIF-scale 1.5-MJ ignition target covering ℓ modes 2:2:200 on a half-sphere using 750-k rays per ray-trace step. Both simulations use the same adaptive fifth-order Runge–Kutta quadrature. The simulations show dramatically different noise characteristics (see Fig. 26). The 2-D random distribution (purple) noise floor overwhelms the modal cutoff at ℓ = 200. A low-noise simulation should show a distinct modal cutoff past the applied signal modes (ℓ = 200 in this case). Noise absolutely corrupts any simulation or other deposition code that cannot show the distinct modal cutoff past the applied signal, and the results cannot be trusted to represent mode growth of the applied signal. In addition, simulations or other deposition codes that apply imprint signal past that supported by the mesh resolution (e.g., at least 10–12 zones per smallest wavelength) tend to compound the numerical noise by confusing noise with signal that cannot be properly supported by the mesh resolution and mask a noisy algorithm by inaccurately interpreting it as true signal. Only the adaptive inverse projection algorithm shows high fidelity where the noise floor is below the applied mode signal. The noise from the 2-D random distribution also affects applied ℓ modes 2:2:200 as compared to the adaptive inverse projection algorithm (red). The random ray distribution has insufficient noise reduction ability, cf. the adaptive inverse projection algorithm.

Another side benefit of using the adaptive inverse projection algorithm is that the expected pristine deposition patterns can be used to diagnose ray-trace problems. Ray-trace codes that use a random distribution induce large of peak-to-peak noise on each time step, making it impossible to determine ray-tracing artifacts resulting from algorithmic problems or assumptions; any potential problems remain hidden, masked and overwhelmed by the noisy deposition. Ray-trace problems are noticed only when a smooth deposition is expected, e.g., via inverse projection, or after it is too late and noise has imprinted in the shell.

The CBET effect is altered by the relative wavelength or angular frequency of the interacting rays, as specified by the wave-vector–matching condition η given in Eq. (4). Knowing the instantaneous wavelength or angular frequency of the interacting rays is an important aspect of CBET simulations. In addition, understanding the scattered-light spectral features aids diagnoses of the experiments with CBET mitigation schemes. Streaked scattered-light measurements record the spectral history of the CBET interaction, and various spectral features signify its activity (see Fig. 6). This appendix describes the Doppler shift in expanding plasmas and explains some observed spectral features.

A light-based ray's vacuum wavelength relates to the angular frequency ω in the usual way, viz.,

The refractive index within a plasma, given by⁴² 

nonlinearly depends on the normalized electron density $ne′,$ both of which are functions of space and time. A field propagating through a medium with a time-varying refractive index becomes temporally phase modulated over the path length L, which alters the characteristic angular frequency, viz.,

(a generalization of the conventional Doppler shift³⁰). The expanding plasma dynamically alters the instantaneous refractive index in space via the electron density Eq. (B2), and thereby the angular frequency Eq. (B3) and wavelength Eq. (B1). This phase-modulation process is independent of the ray-propagation direction.

Taking the temporal derivative of Eq. (B2), a differential equation relating the temporal derivatives of the refractive index n_refr and the electron density $ne′$ is found, viz.,

where the sign determines the shift, e.g., negative is a blue shift and positive is a red shift. An instantaneous wavelength change is commonly referred to as a “chirp” because the angular frequency and wavelength vary as functions of time similar to a bird's song. The frequency chirp is independent of propagation direction, i.e., an inbound ray suffers the same chirp as an outbound ray traversing the same spatial region. The frequency chirp's sign depends not only on an apparent receding or advancing surface as in the common Doppler shift but depends additionally on the sign of the time-varying $∇ne′.$ For example, a receding negative electron density gradient generates a red shift, while a receding positive electron density gradient generates a blue shift; in this way the Doppler shift given in Eq. (B4) is not analogous to the common Doppler shift but they both alter the apparent wavelength. The effect given in Eq. (B4) is best thought of as optical phase modulation to avoid misconception.

Invoking the equation of continuity forms the solution to the temporal derivative of the electron density. In a simulation, the cell-centered value is calculated by solving the volume average of the temporal derivative using the divergence theorem, viz.,

where vol is the averaging volume (typically the computational cell volume) whose bounding surface area is A; v_fluid is the fluid velocity; and $n̂$ is the outward-pointing normal on the boundary surface. The net result of Eq. (B5) demonstrates that the frequency chirp or optical phase modulation suffered by a ray within a cell is solely determined by the volume-averaged temporal derivative of the electron density and not the propagation direction or the direction of any moving surfaces.

OMEGA shot 60000 is used to demonstrate the effects of chirp and electron transport models. The schematic of the target and the pulse used in the DRACO simulations are shown in Fig. 27. If the plasma flow is radial, a graphical interpretation of Eq. (B5) is illustrated in Fig. 28. The balance of the electron density and fluid velocity gradients determine the sign of the frequency chirp as a ray traverses the corona (see Fig. 29). A pulse shape with narrow pickets and a sustained main pulse initiate a predictable sequence of Doppler-shift events as outlined in Fig. 30. Figure 31 (online only) shows a movie describing the sequence coordinated with Fig. 30.

The dispersed frequency modulated bandwidth applied on the Omega laser to smooth the transverse far-field speckle pattern is smaller than the observed chirp but it can be of similar magnitude, e.g., $∼2Å,UV$ on OMEGA, which forms a discrete set of spectral modes and referred to as smoothing by spectral dispersion (SSD). The current levels of SSD on the NIF are much lower, $∼0.4Å,UV$⁠. However, larger bandwidth mFM-SSD⁴⁸ drivers are planned, but the bandwidth will be degraded during the main drive when CBET is most active. Currently, DRACO does not model the wavelength modal spectrum of SSD in connection with CBET; it does model speckle imprint and smoothing. Future modifications will incorporate the simulation of the dynamic modal spectrum of SSD using a similar method as for wavelength detuning, in that, it will model the effects of wavelength separation of the discrete SSD spectrum in conjunction with any wavelength detuning during CBET modeling but not any dampening related to coherence-disruption of LPI growth.

The simulations of shot 60000 are also used to illustrate the combined effect of CBET and the choice of electron transport model (see Fig. 32). The effect of flux-limited (f/0.06) electron transport⁴⁵ is shown in Fig. 32(a). The effect of the nonlocal model based on the Schurtz–Nicolaï–Busquet method¹⁰ is shown in Fig. 32(b). The role of CBET does not change the overall shape of the electron density, but rather it changes the amplitude of its underlying functional form. The electron transport model is responsible for the gross changes in functional form and not the CBET effect, as is sometimes misunderstood.