Scaling laser preheat for MagLIF with the Z-Beamlet laser

The Magnetized Liner Inertial Fusion (MagLIF)¹ concept pioneered at Sandia National Laboratories requires the successful integration of many subsystems including the pulsed power driver for the magnetically driven implosion (Z-accelerator), applied axial magnetic field (10–30 T), and $2ω$ 4 kJ Z-Beamlet laser (ZBL)² to preheat the fuel. The MagLIF target design typically consists of a 4.65 mm inner diameter (ID) Be liner (cylindrical tube) with a 10 mm tall imploding region and with 1.5 mm tall, 3 mm ID cushion features above and below to mitigate the wall instability.^3,4 The target is filled with gaseous D2 fuel at densities between 0.7 and 1.4 mg/cc (60–120 psi at room temperature). The gas is contained at the top of the target by a polyimide laser entrance hole (LEH) foil (made of CHNO), whose thickness is the minimum required to contain the pressure.

Previous works have identified the preheat energies required to optimize the MagLIF target performance for a given fuel density and applied axial magnetic field.^5–7 In general, for low values of preheat energy, the resulting implosion adiabat is too low and does not reach high enough temperatures at stagnation. For large values of preheat energy, the axial magnetic field is advected out of the fuel by the Nernst effect, and this increases energy losses due to thermal conduction. Increasing the applied magnetic field increases the optimum preheat energy because more magnetic field can be advected out of the fuel before the magnetic insulation is compromised. Additionally, increasing the magnetic field also reduces the Nernst effect. Recent experiments⁸ have successfully increased the current delivery (19+ MA), applied magnetic field (16 T), and preheat energy (1 kJ), resulting in higher neutron yields. However, the preheat energy coupled into the fuel remains far below the optimum predicted by simulations at these current and magnetic field parameters. Ultimately, on Z, an increase in the fuel density (1.4–2.0 mg/cc), applied axial magnetic field (30 T), and drive current (>22 MA) can significantly improve MagLIF neutron yields but requires significantly increased preheat energy, which remains a significant challenge on Z today.

There are three main challenges when attempting to successfully laser preheat a MagLIF target. First, the laser energy must be coupled to the implosion region, roughly 1 cm in axial length between the cushions. In order to match the laser propagation length to the target length, there is some optimization of laser power, laser spot, and gas density to produce the required preheat conditions. To successfully accomplish this, the next two challenges must also be considered in tandem.

The second challenge is that the preheating laser beam must first penetrate the LEH foil window, typically between 1.5 and 3 μm thickness for room-temperature targets. Including the window introduces complications to achieving the optimal preheat state for a given target using ZBL. First, the window can be a strong source of laser-plasma-instabilities (LPIs) since the laser must initially interact with a solid density material.^9,10 LPIs can be detrimental to MagLIF as it can backscatter laser energy or redirect it toward metallic surfaces introducing high Z mix.¹¹ Window material can also be directly injected into the fuel both hydrodynamically and kinetically.¹² Finally, the window can absorb substantial fractions of the total laser energy, limiting the total energy that can be coupled to the fuel to approximately 50%–60% of the incident energy in room-temperature experiments.¹³ The complications associated with the laser-window interaction have motivated efforts to mitigate its effect by cryogenically cooling the target, which reduces the gas pressure (and hence required window thickness) for a given gas density. Reducing the window thickness should reduce LPIs and window mix and increase the fraction of delivered energy that is coupled into the fuel. Laser gates⁶ or wire gates¹⁴ are also possible alternatives for completely eliminating loss to the window, but the techniques are still under active development.

The third challenge is related to the presence of the axial magnetic field, which can affect the laser coupling. The primary role of the axial magnetic field is to thermally insulate the fuel from the cold metal liner near stagnation. However, the axial magnetic field is diffused into the fuel before laser preheating and so can affect the laser coupling, as well as be redistributed by the laser deposition. Thermal insulation of the plasma is known to increase the thermal filamentation of the laser since temperature gradients are less effectively smoothed by thermal conduction. This can lead to intensification of the beam and instigate secondary backscatter instabilities. Thermal gradients also drive advection of the magnetic field to colder regions by the Nernst effect. On the nanosecond timescale, since the plasma beta >1, the thermal gradients also drive the hydrodynamic motion of the plasma radially outward, advecting the magnetic field with the flow, and so change the transport properties of the plasma.

In this paper, through 2D and 3D MHD simulations, we assess the ability of the Z-Beamlet laser to preheat cryogenically cooled MagLIF targets for a variety of configurations. Our detailed laser preheat calculations include the two intertwined effects of hydrodynamics and magnetic field on the laser coupling. The main results of the paper are as follows: Sections IV and V show the necessity of increasing the on-target spot diameter of the laser from 1.1 mm to 1.5 mm to reduce thermal self-focusing (TSF) and to more uniformly heat the fuel volume with ZBL's current day-to-day capabilities. Section VI considers higher delivered energies from ZBL, up to 6.3 kJ, which require some upgrades. In, Sec. VII, we assess the impact these laser configurations have on 2D integrated MagLIF performance. Specifically, with an upgraded 6 kJ laser, simulations predict a deuterium (DD) neutron yield of $YDD≃1.1–1.5×1014$ for B_z ranging from 15 to 30 T and 20 MA of drive current. This represents a 10-fold increase in the currently achieved yields for MagLIF. Finally, we present several interesting and unique 3D effects that are observed in the simulations. These observations invite for future investigations on 3D effects on MagLIF, which could be addressed in future publications.

Magnetic fields are well known to affect the heat flux in a plasma and are an important consideration for MagLIF. Magnetized heat flow is parameterized by the electron Hall parameter $xe=ωcτei$⁠, where ω_c is the electron cyclotron frequency and τ_ei is the electron–ion collision time. The parameter x_e scales as $∝Te3/2|B|/ne$⁠, so the initial choice of fuel density and axial magnetic field is an important consideration for the level of magnetization that is achieved at preheat or stagnation. High magnetization of the plasma corresponds to the regime where $xe≫1$⁠. At the MagLIF stagnation, the combination of the compressed density (⁠$ρ∼0.2$ g/cc), compressed magnetic field (⁠$Bz∼5–10$ kT), and temperature (⁠$Te>2$ keV) produce high magnetization conditions in the fuel (⁠$xe>75$⁠). We contrast this with the preheat phase, where no compression of the density or magnetic field has occurred, but the temperature remains high $∼0.5–1.0$ keV, which still produces $xe>1$ during active laser heating. This indicates that significant magnetization effects on thermal transport are still expected, which in turn can be tested by experiments. Additionally, we expect any effect of non-local conduction to be reduced as $RLarmor/λei,mfp<1$⁠, where $RLarmor$ is the Larmor radius and λ_ei is the electron mean free path.

For both the preheat and stagnation phase of MagLIF, the plasma is roughly 10 mm tall, with a characteristic radius for preheat being the laser spot $∼1$ mm and at stagnation $∼0.05$ mm. Assuming that the axial variations are small, the dominant thermal gradient is, then, in the radial direction. For a magnetic field primarily in the z direction and ignoring any diffusion of the azimuthal drive field, the anisotropic heat flux is approximately

For large values of x_e, both $κ⊥∝xe−2$ and $κ∧∝xe−1$ become much smaller than $κ∥$⁠, resulting in significantly reduced radial heat flux, a key requirement for MagLIF at stagnation. For 2D calculations, the heat flux is limited to the rz plane so that azimuthal heat flow through $κ∧$ [Righi–Leduc (RL)] is neglected; however, it is included in 3D. Due to the weaker scaling with x_e, the Righi–Leduc (RL) heat flow can become dominant once the radial conduction is sufficiently quenched. Note that Biermann self-generated azimuthal magnetic fields (⁠$∼∇ne×∇Te$⁠) can modify in plane thermal flux through RL heat flux, but these fields are generally small in magnitude and extent, so typically, the RL term is small. The resulting magnetized heat flow within the laser heated plasma is an important consideration for the development of the filamentation instability described in Subsection II B.

The magnetized transport coefficients depend on the local magnetic field distribution, which in turn is influenced by terms in the generalized Ohm law. The dominating terms are resistive diffusion, advection by the fluid itself (frozen-in flow), and advection with the electron heat flow perpendicular to the local magnetic field direction, known as the Nernst effect. We are primarily concerned with the Nernst term during the laser pulse, as it will tend to advect magnetic flux out radially as the plasma is heated, before the plasma can hydrodynamically respond. This decreases the thermal insulation effect as the laser propagates. To characterize the importance of this effect, let us define the Nernst parameter Ne as the ratio of the Nernst-advection term to the hydrodynamic-advection term of the magnetic field. In the strongly magnetized limit, we have

In terms of useful units, the Nernst parameter is

where we assumed that gradients scale as the laser-spot size $Rspot$ and $Ṙ$ is some characteristic velocity either of the thermal blast wave or of the imploding liner. Thus, it is expected that the Nernst term will become more important as the plasma becomes less conductive and when gradients from laser heating become larger. As we will show, thermal filamentation of the plasma can generate spatial gradients much smaller than the laser spot, which approach the MagLIF stagnation size of $∼50μm$ and substantially increase the Nernst effect in those filaments. After laser heating of the plasma, Nernst advection, as well as thermal conduction, is substantially reduced due to the expansion and subsequent cooling of the plasma as the thermal energy is converted partially to the kinetic energy of the blast wave. At this point, hydrodynamic motion dominates the transport of thermal energy and the magnetic field.

Energy deposition via laser preheat into the MagLIF fuel will largely depend on the laser-propagation depth, the filamentation instability, and the self-focusing effect, as well as the aforementioned transport effects. In the following, we present simple models that will help elucidate the trends observed in the HYDRA simulations, which incorporate all of these processes.

The propagation depth, $z⋆$⁠, of a laser beam absorbed by inverse bremsstrahlung in a plasma was studied in Ref. 15. The result was later re-derived in Refs. 1 and 16–18. In terms of convenient units, $z⋆$ of a deuterium (DD) plasma is

To derive this result, hydrodynamic motion, heat conduction, and LPI were excluded, and the plasma is assumed to be underdense. Some of these nonlinear effects, specifically laser-self-focusing, can substantially modify the laser propagation length as we shall show below. In Eq. (4), $Epreheat$ is the energy deposited by the laser, $Rspot$ is the radius spot size of the laser, $ρDD$ is the DD gas-fill mass density, and the laser harmonic $N=1.054/λμm$⁠, where N = 2 for ZBL. Finally, we have assumed a Coulomb logarithm of $ln Λ=7$⁠, consistent with values from HYDRA simulations. Intuitively, $z⋆$ increases when $Epreheat$ increases, when the laser-spot size $Rspot$ decreases (due to larger intensity), and when the DD fuel density $ρDD$ decreases (due to less absorption). Increasing the frequency to $3ω$ would, then, increase the propagation distance relative to $2ω$⁠.

A prominent effect arising in HYDRA simulations of the MagLIF preheat is the filamentation instability of the laser. This process is important as it may lead to local increases in the laser intensity, causing enhanced stimulated scattering or even additional mixing of impurities when laser energy is deviated toward the liner. In the case of MagLIF, laser filamentation is primarily thermally driven. In the highly magnetized limit, the maximum spatial growth rate of the thermal filamentation instability is given by¹⁹ 

where I is the average laser intensity, n_e is the electron plasma density, T_e is the characteristic electron temperature, and $ncr$ is the critical density. In terms of convenient units,

The general trends for $κfil$ can be understood as follows: Laser filamentation increases with B since higher magnetic fields better thermally insulate the plasma. In consequence, the plasma can support higher local temperature and pressure gradients, which would more rapidly evacuate electrons from the local laser-intensity channels. This leads to more laser filamentation. Similarly, $κfil$ increases with ρ and decreases with T_e since higher fuel densities ρ and lower temperatures T_e make the plasma less conductive. Finally, laser filamentation will be stronger for more intense laser fields. A useful metric that will be used to compare the various cases below is the filamentation growth function $Gfil=κfilz⋆$⁠. This metric considers that the filaments grow throughout a characteristic length defined by the propagation length $z⋆$⁠.

A related effect that also needs to be considered when designing a laser preheat configuration is thermal self-focusing (TSF). Local intensification of the beam, due to some seed, results in higher temperatures (enhanced with magnetic insulation), which both increases the path length of the laser [see Eq. (4)] and leads to the formation of a focusing density channel through subsequent hydrodynamic motion. Additionally, LPIs such as stimulated backscatter can also be exacerbated by the local intensification of the beam. In a highly magnetized plasma, the laser self-focusing length $zSF$ is approximately $zSF=2 π/(4κfil)$ or, equivalently,¹⁹ 

Similar to the filamentation instability, the laser tends to more strongly self-focus as the plasma becomes less conductive. For temperatures ranging from 0.5 to 1.0 keV, with a 15 T field, the self-focus length generally remains below 1 mm, indicating that self-focusing would be observed in all cases. However, as we will show below, there are cases where whole beam self-focusing leads to detrimental effects but also cases where some self-focusing is tolerated and disrupted by filamentation of the beam.

Finally, for our vacuum intensities, we expect ponderomotive filamentation and self-focusing to be less important than similar effects caused by thermal pressure discussed above. In particular, for magnetized cases, the electron gyro-radius is much smaller than the electron mean-free path such that thermal perturbations can quickly grow. Additionally, using typical ponderomotive filamentation and self-focusing figures of merit (FFOM), we find FFOM $∼0.5–0.8$⁠.^20,21 Focusing channels produced by thermal pressure can cause intensification exceeding ponderomotive thresholds; however, we have also run a few HYDRA simulations without ponderomotive effects and found negligible quantitative differences. The major qualitative differences were slight elongation of the filament channels. This suggests that thermal filamentation and self-focusing are the dominant, though we again emphasize that this coupling should be revisited with a more sophisticated LPI code.

The results from fully nonlinear rad-hydro simulations shown in this paper will qualitatively agree with the general trends discussed in this section. Importantly, these effects are also expected to persist to future high yield MagLIF configurations.⁵ The scaling of these effects and other LPI-related effects for plasmas expected in larger pulsed-power drivers is discussed in Ref. 18.

The simulations shown here are run with the HYDRA code.^22–24 HYDRA is a well-known massively parallel arbitrary Lagrangian–Eulerian (ALE) 3D radiation magnetohydrodynamics (RMHD) code. It is routinely used for designing experiments on the National Ignition Facility (NIF) and on the Z machine⁷ and has also been successfully applied to laser preheat experiments in support of MagLIF,²⁵ including other laser systems such as OMEGA-EP.^26,27 Laser energy is deposited through inverse Bremsstrahlung (IB) absorption with HYDRA's laser ray-trace package, which accounts for refraction and reflection of the rays, as well as ponderomotive pressure and momentum transfer. We do not consider any non-linear modifications to the IB absorption for the low Z plasmas here, and laser-plasma interactions such as stimulated backscatter are not considered. These simulations include multi-group radiation transport with non-Local Thermodynamic Equilibrium (NLTE) opacities. While the model choice is less important for deuterium, the radiative cooling rate of the plastic window material can influence how much incident energy is absorbed by it. The window is modeled with sub micrometer resolution in the axial dimension (40 cells) but rapidly expands to follow the explosion of the material. In the radial direction, the resolution is approximately 5 μm. Tabulated equations of state are used for the polyimide windows, deuterium, and aluminum washers. A combination of the Lee–More²⁸ and Epperlein–Haines²⁹ models for magnetized transport coefficients, according to the charge state, Z, of the plasma, are used by HYDRA's generalized Ohm's law discussed next.

Past 2D HYDRA simulations of MagLIF considered only a resistive Ohm law,⁷ including advection by the fluid motion and parallel and perpendicular components of heat flux. More recently, a full 3D generalized Ohm law has been added to HYDRA, which includes effects such as Nernst advection, Biermann self-generated magnetic fields, and full 3D anisotropic thermal conduction. The formulation is described in detail in Ref. 30. While effects, such as Nernst, have primarily been examined in the context of MagLIF stagnation, detailed laser preheat calculations have not included these effects until now. A few simulations throughout were also run with the MHD terms disabled (hereby referred to as “no MHD”) to test their effect in unmagnetized configurations. For all cases, a thermal flux limiter of 0.1 was used. Finally, the Hall term is also present in HYDRA's implementation of the generalized Ohm law. However, densities are typically high (⁠$∼1020$/cc), and the plasma β is also large, which minimizes the impact of Hall physics. As a result, the simulations presented here do not include Hall physics.

A major goal of this work is to understand the laser energy requirements for the Z-Beamlet laser in order to improve and optimize preheat for MagLIF. The HYDRA calculations we show utilize a ray-trace approximation and so are inherently limited compared to a laser-plasma (LP) code that includes effects such as diffraction and backscatter. Diffraction and associated beam spray effects can lead to stalling of the beam, which could alter our assessment of the energy deposited in the implosion height.³¹ Neglecting backscatter implies that these calculations place a lower bound on laser energy requirements. However, our philosophy for designing these configurations has been to continue to utilize relatively low intensities to stay below ponderomotive and backscatter thresholds that have proven successful in modeling past experiments.^13,17,26,27 With this in mind, we anticipate these unmodeled effects to be relatively unimportant in the context of defining the overall laser energy requirements. To better understand the details of the deposition, particularly with magnetic fields, more sophisticated laser-plasma interaction models may be required and will be a focus of future work.

The simulated geometry is shown in Fig. 1(b) and is based on gas cell experiments in the Pecos target chamber at Sandia.⁹ It includes a cylindrical plastic “liner” and end-cap and an aluminum washer that holds a polyimide plastic window. The 15 mm tall simulated volume encompasses a standard MagLIF target whose inner surface is traced inside the gas cell target for reference but not modeled here. Three millimeter diameter cushions at the top of the MagLIF target limit the spot sizes that can be used for laser preheating. The laser propagates from the top of the simulation domain in the $−e→z$ direction and is focused onto and first interacts with the LEH foil, located at 11 mm. The 10 mm imploding portion of a MagLIF target lies between 0 and 10 mm in the simulations. Only the energy deposited within this length participates in effectively preheating the fuel. Laser energy deposited above 10 mm and below 0 mm is wasted to some degree because the fuel in these regions will not participate in the implosion. Furthermore, laser energy that propagates past 0 mm may directly illuminate the bottom cushion feature and introduce mix. For these reasons, it is beneficial for the laser to not penetrate past 0 mm. Energy deposited in the region between the window and top of the imploding height can be transported downward hydrodynamically or through thermal conduction or radiation, but for this work, it will generally be considered as “lost.”

Two laser spot sizes (1.1 and 1.5 mm) are considered in the following simulations, which utilize the spot profile measured at the best focus spot for each distributed phase plate (DPP). The 1.1-mm spot is routinely used in warm MagLIF experiments on Z and Pecos experiments since thinner windows can be used with smaller diameter washers. However, these LEH windows are still relatively thick (∼1600 nm) and can absorb a non-negligible amount of the limited laser energy. The smaller 1.1-mm spot size also has a larger laser intensity (at fixed laser energy), which in turn leads to longer laser propagation lengths and enhanced filamentation and self-focusing. To reduce the energy absorbed by the LEH window and decrease the laser intensity, the target fuel can be cryogenically cooled.³² This allows for a thinner LEH window (500 nm) and for a larger laser spot size (1.5 mm).

It is worth mentioning that the LEH window deforms under pressure. For a cryogenically cooled fuel, we assume a bubble of roughly 60 μm in height, due to reduced fuel pressure; however, in the future, this should be quantified by the measurement. In warm targets, the high gas pressures can lead to bubble heights as high as 600 μm and can subtlety affect the de-focusing of the laser as it burns through the window. Gas densities considered in the following simulations range from 0.7 to 2.0 mg/cc, or 5% to 14% the critical density of $2ω$ light, and span the range considered by scaled designs.^5,7

Thinner windows also allow for a simpler laser pulse strategy to disassemble the window and reduce mix. For warm targets, the so-called co-injection protocol is used¹² where a small pre-pulse of 20 J, from the Z-Petawatt (ZPW) laser, disassembles the window 20 ns ahead of the ZBL heating pulse. A short 1–3 ns lower intensity foot at the start of the ZBL pulse re-heats the radiatively cooled window material before a higher-intensity main pulse heats up the gas. For cryogenic targets with thin windows, it is generally sufficient to use the ZBL alone. This frees ZPW for other applications, such as backlighting or adding additional energy to the main heating pulse. A laser pulse, measured from the ZBL to contain $∼2.6$ kJ, is shown in Fig. 1(a), along with the other higher power pulses used to drive the simulations in this work. The low intensity foot is retained to disassemble the window.

In this section, we describe the simulation results for cryogenic gas cell targets with assumed limitations in the preheat target design recently considered for warm targets.⁸ These amount to a 1.1-mm DPP, 1.0 mg/cc gas fill, $∼2.6$ kJ laser pulse (⁠$I≃7.4×1013$ W/cm²), and maximum 15 T applied magnetic field. We will highlight the limitations of this design space, including important sensitivities to the applied magnetic field, density, and the dimensionality of the system (2D vs 3D). The findings here will motivate the choices made in Secs. V and VI.

We first compare the effect of a 15 T axial magnetic field on laser heating. Figure 2 plots a snapshot of 2D simulations at just over 2 kJ delivered from the pulse in Fig. 1(a). Electron temperature is shown on the left, mass density to the right, and the black line at $z∼11$ mm indicates the window/gas interface. Propagation of the laser was generally uniform, both radially and axially for the 0 T case [Fig. 2(a)]. Although the depth was excessive for a MagLIF target, this can be controlled by truncating the pulse. However, this necessarily limits the energy that can be delivered to the implosion height. At these conditions, Eq. (4) predicts a propagation length of 19 mm, which roughly agrees with Fig. 2(a).

A 15 T axial magnetic field (aligned with the vertical direction) was, then, applied to the simulation. Figure 2(b) plots the result including only resistive diffusion and frozen-in advection (⁠$ηJ=E+v×B$⁠). With the applied field, a very notable temperature increase was observed on axis in the window and gas (⁠$Te>1$ keV), as well as a faster burnthrough time (time for the laser to reach the bottom of the target) due to the high level of magnetization in the plasma. Radially outward motion of the gas, caused by thermal pressure, also creates a focusing channel that helps confine the beam on axis, which we refer to as thermal self-focusing. This effect is enhanced at higher magnetic fields due to increased thermal pressure caused by magnetic insulation. Some of the beam escapes the channel by filamenting radially outwards, but most of the energy is refocused toward and down the axis. With T = 700 eV as the characteristic temperature of the plasma, Eq. (6) gives $κfil≃1.9$ mm⁻¹ for the filamentation spatial growth rate. It is, therefore, not surprising to see the development of filaments and self-focusing at these preheat conditions.

Utilizing a characteristic velocity from simulations of $Ṙ≃100$ km/s, Eq. (2) yields $Ne≃10%$⁠, suggesting that the Nernst term should be important. Figure 2(c) plots the result of the simulation with a 15 T axial magnetic field with the Nernst term included. The result shows more radially diverging filamentation of the beam. Energy striking the bottom of the simulation is reduced by only 100 J, indicating that most of the energy is still focused along the axis. Nernst advection also clearly influences the redistribution of the magnetic field. Figure 2(d) plots $ωcτei$ for the Nernst (left) and no-Nernst (right) cases. Without the Nernst term, nearly the full height of the plasma is well magnetized and thermally insulated due to the high temperatures. Only the top of the target indicates some advection of the field due to the fluid motion. With the Nernst term included, large radial thermal gradients advect flux to the periphery of the plasma, which affects the uniformity of the plasma magnetization. Magnetic flux can be subsequently swept up by the blast wave and lost through diffusion to the MagLIF liner during the implosion. From these results, including Nernst is more important for understanding the magnetic field distribution in MagLIF rather than strongly affecting the laser energy coupled.

Equation 4 shows that higher density reduces the propagation depth, which would improve the energy coupled to 10 mm, but comes at the expense of a shorter self-focus length and increased filamentation growth rate [Eqs. (6) and (7)]. Due to the severe self-focusing observed in the 1.0 mg/cc with 15 T case, improvement would not be expected at higher density. This was confirmed in the results of 2D simulations run with 1.4 mg/cc gas and 15 T, which also showed propagation dominated by severe self-focusing down the axis. In fact, as shown in Table I, the 15 T, 1.4 mg/cc case actually coupled 140 J less to the implosion height than the 1.0 mg/cc case due to TSF. Without an axial magnetic field, self-focusing was substantially reduced, resulting in 180 J more coupled to the imploding height and better matching the expected propagation depth (⁠$∼12$ mm) from Eq. (4). Although considerably less expensive than 3D simulations, computations done in 2D appear to enhance whole beam self-focusing and can also artificially constrain the laser to filament into “rings” in the $rθ$ plane. To demonstrate this, we next show results from a few 3D simulations.

The 3D simulations were performed with 1.0 and 1.4 mg/cc fuel, using nearly identical inputs as the 2D calculations, but with the full 3D laser package and generalized Ohm law.³³ Slices through y = 0 of the 3D simulations are shown in Fig. 3 at the same time as the 2D calculations for direct comparison. One can immediately notice that the 3D simulations show less radial narrowing of the beam deeper into the target that drives energy out of the implosion region. Instead of continued whole beam thermal self-focusing, observed in 2D, the beam instead filaments and expands out radially, shortening the propagation depth. Figures 3(d) and 3(e) with 1.4 mg/cc also show that, as the field strength is increased, filamentation of the beam also increases. This remains consistent with faster thermal filamentation growth due to the reduced thermal conductivity of the plasma. For the higher field cases, the evolution into filamentation tends to increase the deposited energy within 10 mm. However, this comes at the expense of more asymmetric and random propagation that directs the laser toward the liner walls.

For the no-MHD cases in Figs. 3(a) and 3(c), there is only minor TSF at $z≈6$ mm where there is a modest increase in T_e. For the magnetized cases, the self-focusing around $z=6$ mm becomes more obvious, identified by the hot spots in the plasma, which are now well insulated and much hotter. Clearly, self-focusing persists in 3D, but the peak intensity in this region is reduced by roughly a factor of 2–3, compared to 2D. Unfortunately the intensity remains exceedingly high, ranging from $6×1014$ to $1.5×1015$ W/cm², which poses a large risk of backscatter and truncating propagation.

Table I summarizes the results obtained for the energy deposition into the imploding fuel region between 0 and 10 mm of the target. Generally, the 3D simulations show increased coupled energy with B_z as a result of substantially more thermal filamentation that disrupts and overwhelms any additional whole beam self-focusing, which appears to be an artifact of 2D. Nonetheless, the propagation depth of the laser in 1.0 mg/cc is still nearly the full implosion height and ultimately limits the coupled energy, which is notably below optimal preheat values discussed here and in Refs. 5 and 6. Losses from stimulated Raman scattering (SRS) or stimulated Brillouin scattering (SBS) are also likely to degrade the energies tallied in Table I for both 2D and 3D scenarios, further limiting the energy coupled by this spot size. Finally, for higher field levels, the erratic filamentation and propagation of the beam also become concerning, particularly for the 1.4 mg/cc case where this occurs higher in the target.

These issues with the 1.1-mm DPP can be overcome with a larger spot size at the expense of losing more energy to the window as the illuminated area increases and necessitates the use of thinner laser entrance windows. The remainder of this paper focuses on the benefits for MagLIF, which can be achieved with a larger laser spot size, in particular a 1.5-mm DPP that has recently been made available at Sandia.

From Eq. (4), the propagation depth and self-focusing are expected to be reduced by the increase in spot size (via reduction in intensity to $∼4×1013$ W/cm²) relative to the 1.1-mm DPP for fixed laser power and magnetic field strength. However, the theory does not provide a clear answer as to when nonlinear whole beam self-focusing takes over as was observed with the 1.1-mm DPP. Here, we fix the fuel density at 1.0 mg/cc and scan the magnetic field with the 1.5-mm spot to show the efficacy of the 1.5-mm spot at mitigating the nonlinear effects.

The results of these 2D simulations are shown in Fig. 4, which depict the electron temperature and mass density at just above 2 kJ delivered for direct comparison with Fig. 2. There are noticeable changes in the filamentation of the beam as the axial magnetic field strength is increased, from 0 to 50 T; however, there is very limited TSF down the axis in all cases. Thus, at these intensities, thermal filamentation of the beam, enhanced by the thermal insulation from the magnetic field, is the dominant effect. For $Bz=0$ T, filamentation is reduced due to the smoothing effect of thermal conduction, and due to refraction, there is a slight narrowing of the heating deeper into the gas. As the field strength increases, the thermal insulation increases from $κ⊥/κ∥∼1/30$ to $∼1/300$⁠. This moves the location at which the heated radius is constricted higher up in the target to $z∼9$ mm. After the constriction, the beam de-focuses and filaments, generating a radial structure in the plasma. Peak temperatures in the plasma also do not substantially increase from 15 to 50 T, indicating a saturation and equilibration between the thermal insulation, collisional absorption, and radiative cooling.

Despite the qualitative differences in filamentation and plasma temperature between the varying field levels, the application of the applied magnetic field does not strongly affect the laser energy coupled to the fuel or propagation depth due to the reduction in thermal self-focusing. The energy deposited as a function of B_z is shown in Fig. 5 and includes the 3D results shown in Sec. V B. Slightly offset to the left of $Bz=0$ T is the “no MHD” case. Overall, the 2D and 3D calculations are in excellent agreement. The increase in coupled energy for 3D is due to lower loss to the window (⁠$∼20$ J) and more thermal filamentation that again stalls the beam in the 10 mm region. The increase in coupled energy as a function of B_z for the full height is due to reduced losses to the window.

These results suggest that for larger aperture phase plates, producing lower intensities, it is possible to use 2D simulations to quickly estimate the energy coupled into the fuel, instead of the more expensive 3D simulations. However, there are remaining 3D features that cannot be captured by 2D simulations that may be important to MagLIF beyond simply the energy coupled to the fuel that we briefly discuss next.

In Sec. V A, we showed that the 2D and 3D simulations produced comparable coupled energies as a function of B_z, which also indicates that the propagation depth is in good agreement. However, there are remaining notable differences in how the laser propagates between the 2D and 3D cases. To highlight these differences, Fig. 6 plots the electron temperature and mass density of 3D simulations, with a $90°$ wedge removed, at the same time as the 2D simulations (roughly 2 kJ delivered). As the beam initially moves into the gas, the heating becomes radially constricted by the axial magnetic field insulation as was observed in 2D. However, as the beam defocuses into the plasma, the filamentation becomes highly 3D, breaking the beam into many individual filament structures that in many cases diverge out radially. Expansion of the beam, then, results in stalling of the propagation in the axial direction, and this effect increases with applied field strength. In the absence of thermal filamentation, it would otherwise be expected that the higher field cases propagate more deeply due to inhibited radial heat flow.

Thermal filamentation in 3D arises from similar sources as in 2D such as intensity modulations from spottiness of the beam or channeling caused by density gradients in the exploded window material. However, in 2D, only a few ring-like channels are able to form, which is sharply contrasted by the large number of filaments produced in 3D. Hot spots that develop can be quenched by thermal conduction, but when the applied field is added, the large reduction in thermal conductivity leads to a feedback mechanism that instead reinforces the hot spots. To illustrate this, Fig. 7 shows $rθ$ slices through the 3D simulations from Fig. 6 for various field levels. As the magnetic field is increased, hot spots become better insulated and more numerous. As the temperature and pressure increase, the plasma hydrodynamically responds and the density drops, further focusing the laser through the filament.

The applied magnetic field plays a large role in enhancing the filamentation, but the filamentation also affects the distribution of the magnetic field in the plasma as shown in the bottom row of Fig. 7. Higher temperature regions lose magnetic flux, via Nernst advection, into cooler and denser regions such that the field accumulates in pockets. These magnetic islands become particularly obvious at 30 T. In 2D, these are annular pockets, and in 3D, these pockets are distributed throughout the heated volume. Since the Nernst effect is the strongest near steep gradients (edge of the circular filament), enough flux is retained (⁠$∼50%$⁠) such that the filaments remain magnetized. The mini-blast waves from the filaments also sweep up the axial field and can push it into the cooler, denser pockets. While some field is certainly advected out at a large radius with the blast wave, a 3D distribution remains in the rarefied gas. Thus, filamentation disrupts direct radial Nernst advection of flux to the blast wave. These effects also introduce azimuthal and radial non-uniformities in the temperature and magnetic field as the blast wave develops, leading to asymmetries in these quantities that persist throughout the implosion.

Investigating the detailed impact of these 3D effects on MagLIF stagnation remains outside the scope of this work and is a focus of current studies. However, it remains important to note that these 3D simulations show some qualitatively different behavior that cannot be captured in 2D simulations, despite making little difference to the expected laser coupling.

In the following, we consider the possibilities for laser preheat using the Z-Beamlet laser assuming upgrades to the available laser energy to achieve optimal preheat energies for MagLIF.

As designed, the ZBL is capable of 4 kJ of output that is degraded by transmission through various optics in the chain to bring the laser to the MagLIF target. Upgrades to optical coatings to improve damage thresholds could allow more routine operation at this energy. For the following simulations, the peak laser power of the previous pulse shape is multiplied by 1.45, producing a roughly 1 TW pulse, while the foot is left unmodified [the yellow trace in Fig. 1(a)]. The total laser energy into the simulation is just over 3.7 kJ—a 1.1 kJ increase. To achieve this in reality, the transmission fraction to the target needs to be 92.5% for 4 kJ output from the ZBL.

Figures 8(a) and 8(c) plot 2D results of the electron temperature for the 1.0 and 1.4 mg/cc cases, with 15 and 30 T at the end of the higher power laser pulse. Comparable 3D simulations with $Bz=30$ T are shown in Figs. 8(b) and 8(d). Slices are taken through y = 0, and the mass density and the electron temperature overlaid. Despite the increased intensity, dramatic whole beam self-focusing remains mitigated with the larger laser spot size, but there is a small increase in propagation depth going from 15 to 30 T. This increases the overshoot energy from 100 to 200 J for the 1.0 mg/cc fill. With the majority of the full volume heated, it is, then, expected that longer pulses will couple less efficiently to the imploding volume. At 1.4 mg/cc, there is only a small overshoot of 16 J for 30 T. The 3D simulations show that thermal filamentation remains the dominant process. Propagation for the 3D simulations lags the 2D simulations slightly due to more energy being lost to the window from the shorter foot pulse used (⁠$∼100–150$ J) but otherwise shows good agreement. Tabulated energies for the 3D simulations are shown in Table III in Sec. VI B.

The total coupled energies for these configurations in 2D are tabulated in Table II. The efficiency of laser energy coupled to the gas (⁠$Edep/Edel$⁠) is $∼72%$ for 10 mm and 80% for the total heated volume of gas. All configurations generally couple $∼2.7$ kJ in 10 mm, of $∼3$ kJ total in the gas, and losses to the window increase by roughly 100 J. The energy deposited above 10 mm accounts for 200 and 335 J for the 1.0 and 1.4 mg/cc cases, respectively. With window mix predicted to be negligible, energy above the imploding region can, in principle, be recovered by reducing the standoff between the position of the window relative to the top of the imploding region. Since more energy is deposited near the top of the target than bottom, this is a potential strategy to maximize the coupling efficiency of the laser.

These results suggest that 2.7 kJ is roughly the upper limit of energy that can be coupled to a MagLIF target using the ZBL as it mostly exists today. However, this still represents an increase of between 0.7 and 1.0 kJ deposited into the fuel compared to warm configurations with the 1.1-mm DPP.

The total available energy that can be output from the ZBL is estimated to be roughly 6 kJ assuming the state of the art fluences from the NIF. Upgrades required include (1) adding laser glass, (2) redesigned spatial filter pinholes in the cavity of the main amplifiers, and (3) improvements to the regenerative amplifier to allow longer pulse lengths. The Z-Petawatt laser may also be added to the ZBL pulse providing an additional 0.8–1.0 kJ. Potential losses in the optical chain in delivering this energy to the target will not be explicitly included. For the sake of conciseness, in this section, we present results from 3D HYDRA calculations. The results from the 2D calculations agreed with the 3D results presented here.

To further increase the total laser energy delivered to the target, the laser power or pulse length must be increased. It is not anticipated that the ZBL power can be raised much higher due to damage potential to the optics caused by focus degradation. Therefore, we consider a fixed 1.0 TW pulse extended in time, which includes a 1 ns 200 J foot [shown in Fig. 1(a)]. Calculations were run for 1.0, 1.4, and 2.0 mg/cc fuel densities and for $Bz=0,15,30$ T. Coupled energy as a function of laser energy delivered to the target, from these nine simulations, is plotted in Fig. 9, where the solid curves show the energy coupled in 10 mm and the dashed curves show the total coupled energy to the gas in the simulation domain (1.4 mm). As mentioned in Sec. VI A, the 3D calculations generally agree with the 2D versions at various delivered energies to within 100–200 J, with early time discrepancies attributable to differences in window losses.

For the 1.0 mg/cc case shown in Fig. 9(a), the coupling to 10 mm starts to fall away from the total coupling as the laser overpropagates. Additional deposition in the implosion region is mainly a result of the laser keeping the already heated material in the implosion region transparent and not heating any new gas. Past 4 kJ delivered, the applied magnetic field does have a detrimental effect on the coupled energy for 1.0 mg/cc by causing some over propagation even in 3D. However, the primary degradation in coupling efficiency is due to the limited stopping power of the gas. For this configuration, the maximum achievable energy that can be delivered is 3.5 kJ.

For the 1.4 mg/cc case shown in Fig. 9(a), the 10-mm energy coupling to the fuel behaves almost linearly to the total energy delivered into the gas. Note that there is also a slight overpropagation of the laser near the end of the pulse for the 15 and 30 T cases. However, at this point, the majority of the heating of the implosion volume has successfully occurred. For this gas fill density, the achievable energy deposition is $∼4.8$ kJ.

The energy deposition for the 2.0 mg/cc case is shown in Fig. 9(c). The higher gas fill density can successfully absorb most of the energy that is delivered to the target. However, due to the higher density conditions, filamentation of the beam becomes much more severe compared to the 1.4 mg/cc case. Figure 10 plots the mass density for 1.4 and 2.0 mg/cc with 0 T (left) and 30 T (right). This is particularly problematic for the 30 T case where a few large filaments propagating at erratic angles carry most of the beam energy. These filaments are likely to be stochastically oriented and introduce a substantial risk of laser interaction with the MagLIF liner. Additionally, the deposited energy is only about 500 J higher than the 1.4 mg/cc case, which means that the specific energy (kJ/mg) of the 2.0 mg/cc fuel is lower than that for the 1.4 mg/cc case. This is not ideal since higher specific preheat energies generally tend to lead to higher fuel temperatures at stagnation and therefore better performance. Therefore, while large amounts of energy can be coupled to the fuel, a 2.0 mg/cc fill does not appear to be an optimal configuration under the assumed constraints.

Finally, we comment on the temporal loss of energy to the thin 0.5 μm LEH windows. Figure 11 plots the energy deposited in the window material as a function of laser energy delivered to the target for the nine 3D simulations just described. For all cases, loss to the window slows down substantially by 3 kJ delivered where it is within about 10% of the total loss and is relatively independent of density. Conductive cooling and radiative cooling of the window material cause the laser to continue to deposit energy in the material throughout the pulse. This shows that the balance of coupled energy between the gas and window is most sensitive early on. As the delivered energy increases, time also increases, allowing the window material to hydrodynamically clear from the laser spot. Because the majority of the losses occur during a relatively early and short period of time, this explains the sensitivity of the energy coupled to the gas to the length and energy of the foot pulse. Thus, while there is an initially steep investment to heat the window, there is some advantage to longer pulses for these high input energies, as the fraction loss to the window dramatically decreases in time.

For convenience, we have tabulated the deposited energies for the three ZBL energy levels (2.6, 3.7, and 6.3 kJ) in Table III for the 3D simulations in this section. These results represent our best understanding of potential preheat energies available for MagLIF using a cryogenic platform with the 1.5 mm DPP. We found that a 1.4 mg/cc fill is the best matched to the MagLIF geometry in terms of maximizing the efficiency of coupling and avoiding excessive self-focusing. Additionally, the deposited specific energy (⁠$Edep/mgas$⁠) is comparable with the 1.0 mg/cc fill and much higher than that of the 2.0 mg/cc fill. In the remainder of this paper, we place the results of these simulations in the context of integrated MagLIF performance.

We now present the results of clean 2D HYDRA simulations illustrating the performance space that can be achieved in current MagLIF targets using the laser configurations described in this paper. These results complement the work by Slutz et al.^5,6 by including the full laser package model of the preheat and the work by Sefkow et al.⁷ by including the generalized Ohm law effects on preheat, as well as stagnation, for some fixed conditions. The only experimental upgrades to MagLIF assumed here are the laser energy and the ability to reach 30 T. The other inputs, 15 T and 20-MA drive current, are achievable with present-day technology. Dimensions of the MagLIF target are also fixed with an inner radius of $Rin=2.325$ mm and with a standard aspect ratio $AR≐Rout/(Rrout−Rin)=6$ beryllium liner. The cushion and window radius is 1.5 mm, and the window thickness remains 0.5 μm. Simulations use the same 1.5-mm DPP and the highest energy laser pulse shape (200-J, 1-ns foot, and a 1-TW heating pulse). The pulse length was varied from 1 to 7 ns in order to deliver between 1.2 and 7.2 kJ.

As a cautionary note, the radial resolution in these integrated simulations is less than the laser only calculations described previously and so produces slightly different coupled energies for a given input energy due to changes in the amount of filamentation and self-focusing. For example, self-focusing can be more pronounced in lower resolution simulations since finer scale filamentation that can disrupt the focusing is not resolved. Additionally, instabilities that affect the liner integrity such as electrothermal^34–38 and magneto-Rayleigh–Taylor (MRT) instabilities^39–42 are not included.

Figure 12 shows expected DD neutron yields for MagLIF as a function of preheat energy deposited with the 1.5-mm DPP. The trends in yield with preheat energy show a similar behavior to that found in Slutz et al.⁶ At low energies, the fuel adiabat is too low to reach fusion temperatures at stagnation, resulting in a “yield cliff” where reductions in the preheat energy result in lower yields. At high energies, the larger temperature and density gradients produced in the fuel advect the magnetic field out of the fuel volume by the Nernst effect, reducing magnetic insulation, resulting in lower yields with higher preheat energies. For reference, the warm 1.1-mm DPP co-injection configuration presently used in MagLIF generally couples between 1.0 and 1.6 kJ to the gas (of 1.8–2.6 kJ delivered to the target) and puts MagLIF performance on or near a yield cliff at these deposited energies for 1.0 mg/cc fuel and well below optimal for 1.4-mg/cc fuel. Operating on a cliff is certainly not ideal. Since we do not account for degradation mechanisms such in flight instabilities or mix, any sensitivity to these effects could lead to less predictable and more irreproducible target performance. With a maximum coupled energy of roughly 2 kJ for the 1.1-mm DPP, MagLIF yield potential is restricted, and potentially more unreliable, without the laser configurations shown here.

As shown in the top of Fig. 12, for the 1.0-mg/cc 15-T case, the 3.7-kJ laser pulse is sufficient to reach a significant yield plateau due to Nernst advection losses and could test this physics. At 30 T, this pulse can nearly optimize the yield performance at 1.0 mg/cc. Reaching the true optimum requires a > 5 kJ pulse due to the overpropagation observed at 1.0 mg/cc. However, at this laser energy, MagLIF would benefit more from a higher density configuration.

At 1.4-mg/cc gas fill, the yield steadily improves for both 15 and 30 T as the deposited energy increases. Increasing the laser energy from 2.6 to 3.7 kJ benefits the 30 T case the most as lower preheat configurations typically produce lower β plasmas at stagnation. The 6.3-kJ pulse reaches the optimal yields for both field levels and also optimizes the overall MagLIF performance at 20 MA when comparing peak yields with the other fuel densities.

Finally, as shown in Sec. VII, the 6.3-kJ laser pulse is insufficient to fully heat the fuel volume for the 2.0-mg/cc case. This reduces the overall specific energy of the fuel for the fixed implosion height and reduces overall performance due to the lower stagnation temperature. To remedy this, either more PdV work must be supplied (more drive current), which will tend to increase the convergence ratio [CR ≡ R_inner(t = 0)/R_inner(t)], or the specific fuel energy raised by increased preheat energy. This is particularly apparent compared to an artificial case using a $3ω$ beam, which propagates deeper and heats a larger volume and increases the yield by 50%–100% at 15 T for the same deposited laser energy in the imploding height. Due to the thermal insulation of the 30-T field, the laser is also able to heat a larger volume but introduces a large amount of filamentation and associated risks [see Fig. 9(c)].

From the results shown in Fig. 12 and the considered initial conditions, the peak yield is $YDD≃1.5×1014$⁠, or roughly 50 kJ of DT equivalent yield utilizing a 20-MA current drive, 4.25-kJ preheat, 1.4-mg/cc gas fill, and 30 T. From the 3D preheat only results, 4.25 kJ is deposited at roughly 5.5 kJ delivered which allows some margin relative to the potential available laser energy. Similar to Slutz et al., the range of preheat energies sample roughly a factor 10 in yield, which is helpful for scaling studies. Finally, higher fuel densities with sufficient preheat are also appealing because they produce lower CR implosions (⁠$∼30$ compared to >40 for current configurations), which may reduce overall instability growth and improve the robustness of the experiment.

To optimize MagLIF performance and define laser needs, we have relied on the general result presented throughout this paper: across a range of delivered energies with the 1.5 mm DPP, we have found that achievable preheat energies with the ZBL are not strongly dependent upon the strength of the applied magnetic field or even the dimensionality of the simulation. The main difference between 2D and 3D is the degree and character of thermal filamentation that produce uniquely 3D effects. We conclude this section by discussing some observations on these effects in idealized 3D simulations.

The first concern, which is difficult to address with HYDRA, is the impact of any mix introduced by the laser striking the lower MagLIF cushions or even the liner itself due to filamentation. In all high energy cases here, the beam expands sufficiently to intersect with the cushions and liberate the surface material. For example, in the integrated 3D simulation discussed next, the bottom cushion absorbs almost 500 J for the 6.3 kJ laser pulse. Generally, HYDRA predicts that any material liberated from these cushions is ejected hydrodynamically downwards, along with fuel mass, as the liner implodes and end losses begin. Any sort of ballistic injection of higher Z material would not be accounted for. Anti-mix layers may need to be considered for these scenarios but can, in principle, be designed with HYDRA.

It was shown in 2D (⁠$r,θ$⁠) simulations by Sefkow et al.⁷ that a radially non-uniform blast wave could be produced by an irregular un-smoothed beam. Here, we have shown that thermal filamentation of the beam is sufficient to produce a non-uniform blast wave. To understand potential consequences to MagLIF from this effect, a fully integrated 3D MagLIF simulation, without significant MRT,⁴³ was carried out with 1.4 mg/cc, 30 T, and the 6 kJ laser configuration, using the same 20 MA drive current as the 2D simulations. Axial slices through the target center are shown in Fig. 13 for T_i and B_z at CR $=1$ and CR $=15$⁠. The top frames show the blast wave just as it reflects from the liner wall (circular thick black line) and the second frame a few ns before stagnation. Comparing the liner/gas interface between the two times shows a deformed liner wall at CR $=15$⁠, which can further grow due to deceleration Rayleigh–Taylor instabilities. This deformation appears to be initially seeded by the impact of the blast wave due to the lack of MRT growth and subsequent feedthrough.

The other apparent feature at CR $=15$ is the presence of strong vorticity in the fuel, denoted by the flow arrows. While asymmetry in the flow following preheat is initially small, the subsequent compression of the fuel by the liner amplifies the deposited vorticity. This produces localized vortices in the fuel core (⁠$∼10$ km/s), as well as high azimuthal velocity shear near the wall near stagnation. The velocity shear can lead to Kelvin–Helmholtz (KH) instability, and the vortices in the fuel can disrupt a uniform stagnation. Resolution in these 3D simulations is insufficient to model the growth of KH or the deceleration instabilities in detail and so cannot predict their true impact. Figure 13(b) also shows that the axial magnetic field is entrained in the flow and not simply advected straight to the walls by Nernst. This also leaves the thermally insulated regions distorted and so can be responsible for some 3D features in stagnation imaging. Viscosity could damp many of these features, but a magnetized treatment is not currently available. However, these 3D HYDRA simulations predict that, without sufficient viscous damping, significant azimuthal flows can be produced in the MagLIF stagnation, which are known to affect laser driven inertial confinement fusion (ICF) capsules^44–47 and wire array Z-pinches.^48,49

The source of these effects, thermal filamentation of the beam, is still present with the 1.1 mm DPP. This suggests that these effects could be present in recent MagLIF experiments and could be observed though their ultimate impact remains unknown. Since physical viscosity scales as $Ti5/2$⁠, these lower energy configurations may also be more susceptible to vorticity effects. There are potential solutions to reducing the thermal filamentation, such as through well-known beam smoothing techniques such as polarization smoothing and smoothing by spectral dispersion (SSD),^31,50 or converting the ZBL from $2ω$ to $3ω$⁠, but they have not been studied in detail with high magnetic fields, which exacerbate the effect. However, if they are necessary, this would, then, potentially place additional costs on ZBL upgrades to achieve the desired goals for MagLIF. This motivates additional work to understand the relative importance of the final fuel structure due to MRT feedthrough and deceleration instabilities and fuel structure generated by the laser. Nonetheless, we have shown that the necessary coupled energies are otherwise possible.

The computational results presented here, including the effect of the axial magnetic field, have identified potential laser preheat energies that could be available for MagLIF using the Z-Beamlet laser. Incident laser energies ranged from 2.6 to 6.3 kJ, spanning current capabilities of the ZBL to future upgrades. The simulations are well resolved, with $∼5 μ$ m radial resolution, but do not resolve speckles.

Both 2D and 3D simulations of the 1.1 mm DPP, routinely used in warm MagLIF experiments, showed that preheat energies on the order of 2 kJ could be achieved. However, due to substantial overpropagation and thermal self-focusing of the beam, it appears unlikely that the significantly higher preheat energies required for optimal MagLIF operation can be reached with the 1.1 mm spot.

With cryogenic cooling, thinner LEH windows, along with larger spot sizes, can be used. A 1.5 mm DPP, already available at Sandia, was used for the remainder of the simulations and provides numerous benefits. The reduction in intensity helps improve the validity of the model by decreasing stimulated backscatter, which is not accounted for in these simulations. For magnetized cases, the lower intensity also mitigated thermal self-focusing, allowing more plasma volume to be heated by the beam. Thermal filamentation and breakup of the beam increased with the applied magnetic field strength but did not strongly affect the coupled energy. While Nernst advection also did not strongly affect the coupled energy, it does cause additional magnetic flux redistribution, not captured by frozen-in advection, which remains important to include in stagnation calculations. As shown in the B_z scan (Sec. V A), these results are sensitive to the degree of thermal insulation in the plasma. As such, they rely on accurate transport coefficients, which are relatively untested by experiment, motivating future magnetized laser preheat experiments.

For the 1.5 mm DPP, 2D and 3D simulations were in general agreement for the energy coupled to the fuel across the range of laser energies delivered to the simulation, as well as over a variety of densities and applied field strengths. The major difference between the 2D and 3D results was the character of the thermal filamentation. In 3D, the beam was more prone to defocusing into radially diverging filaments (outside of the original spot size) that pose the risk of illuminating the liner or cushion. Azimuthal variations in the beam were also amplified by thermal filamentation, leading to a radially non-uniform blast wave. 3D integrated simulations suggest that this can seed deceleration instabilities and vorticity in the fuel. Deposited vorticity is amplified as the liner compresses and ultimately creates velocity shear at the liner wall and also significantly alters the axial field distribution in the fuel due to the residual azimuthal flow. Understanding the impact of vorticity in the fuel on MagLIF performance is now a focus of current computational and theoretical research and could motivate incorporating additional smoothing techniques for the ZBL or frequency tripling. Future calculations can include kinematic viscosity, but modifications due to the magnetic field are not yet available and so limit their utility.

Focused experiments may also show that our hydrodynamic/ray-trace model is insufficient to correctly model filamentation, particularly in our magnetized plasmas where the gyro-radius can be on the order of the speckle size. In the absence of backscatter, speckle level filamentation and spray should further stall the beam in the plasma, which would essentially increase the coupled energy in the implosion height. With the above fact in mind, we emphasize that these results are likely to set a lower bound on laser requirements. In the future, we plan to better quantify this with a suitable laser-plasma code that includes magnetized effects. While we do not anticipate this to dramatically alter the laser energy requirements, the calculations may, for example, suggest additional smoothing protocols if beam spray poses a risk for introducing wall mix. This could also affect the redistribution of the magnetic field in the fuel and associated implications, but incorporating those details into a 3D HYDRA implosion represents a significant challenge.

Finally, 2D integrated MagLIF simulations, with 20 MA drive current, were run with the cryogenic configuration to assess expected performance trends for the various incident laser energies. At 3.7 kJ delivered, yield flattening due to the Nernst effect can be examined at 1.0 mg/cc, 15 T. At 30 T, the preheat is nearly optimal for 1.0 mg/cc. Upgrading the laser to 6 kJ or more was not found to improve target performance substantially for 1.0 mg/cc but could help test key physics of MagLIF such as the Nernst effect at 30 T. However, the upgrade significantly benefits higher densities, producing an overall peak yield with 1.4 mg/cc fuel, 30 T and 4.25 kJ deposited energy producing 50 kJ of DT yield but of course neglect 3D effects, most notably degradation from magneto-Rayleigh–Taylor instabilities. These simulation results show that there are relatively straightforward paths to high preheat energy configurations for MagLIF using a cryogenic platform, particularly when considering ZBL upgrades. Initial experiments for unmagnetized cryogenic laser-only targets are currently planned to test ZBL coupling at the 2.6 kJ delivered level, which will improve our confidence in these models, but should be followed up in the future with the applied magnetic field included.

The authors thank the Z-Beamlet team for helpful discussions on laser capabilities, J. Schwarz for the 1.5 mm DPP profile, M. E. Glinsky, E. P. Yu, and F. W. Doss for insightful discussions, S. A. Slutz and M. R. Gomez for detailed comments on the paper, J. M. Koning and M. M. Marinak for HYDRA code support, and the High Performance Computing (HPC) team at Sandia for computing support. Most figures are produced with VisIt.⁵¹ VisIt is supported by the Department of Energy with funding from the Advanced Simulation and Computing Program and the Scientific Discovery through Advanced Computing Program. One of the authors (D.E.R.) was supported in part by an appointment to the Sandia National Laboratories Truman Fellowship in National Security Science and Engineering, which is part of the Laboratory Directed Research and Development (LDRD) Program (Project No. 209289). This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the U.S. Government. Sandia National Laboratories is a multimission laboratory managed and operated by the National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-NA0003525.

The data that support the findings of this study are available within the article and the references cited herein.