Direct-drive inertial confinement fusion: A review

It has been 55 years since the demonstration of the laser and the first proposal to use focused laser light to initiate thermonuclear (TN) fusion. Understandable optimism led to early hopes and claims that breakeven (fusion energy out greater than laser energy in) was only a few years away. However, it is only in recent years that laser technology and understanding of the related physics have advanced to the point that laser-driven fusion energy can be seen as a realistic possibility for the future.

Inertial confinement fusion (ICF) is distinguished from magnetic confinement fusion in that the fusion fuel is compressed and maintained (briefly) at fusion densities and temperatures by its own inertia. There are two approaches to laser-driven ICF: direct drive, in which a spherical target containing fusion fuel is directly irradiated by laser beams,¹ and indirect drive, in which the laser beams heat the inside of a typically cylindrical enclosure known as a hohlraum, producing x rays that irradiate a spherical fuel-containing capsule.^2,3 This review is concerned solely with direct drive.

This review traces the history of direct-drive ICF from the earliest days to recent years, demonstrating the evolution of scientific understanding in a large number of areas critical to the success of the concept.

Target physics is the primary focus of this review. It is important to recognize that the growth of target-physics understanding has depended heavily on parallel developments in laser technology, experimental diagnostics, computer codes, and target fabrication, some of which have arisen from the indirect-drive program. However, descriptions of these areas are, with occasional exceptions, outside the scope of this review. It is assumed that the interested reader will refer to the bibliographies of the target-physics papers cited here for the pertinent information.

The evolution of the capabilities of high-power laser systems over the period of this review is remarkable. Multibeam spherical implosion facilities include the 20-beam Shiva,⁴ the 10-beam Nova,⁵ and the 192-beam National Ignition Facility (NIF)⁶ lasers at Lawrence Livermore National Laboratory (LLNL); the 4-beam DELTA,⁷ the 6-beam ZETA,⁸ the 24-beam OMEGA,⁹ and the 60-beam OMEGA¹⁰ lasers at the Laboratory for Laser Energetics (LLE) at the University of Rochester; the 8-beam Helios laser¹¹ at Los Alamos National Laboratory (LANL); the 4-beam GEKKO IV¹² and the 12-beam GEKKO XII¹³ lasers at the Institute for Laser Engineering at Osaka, Japan; the 6-beam Vulcan laser¹⁴ at Rutherford Appleton Laboratory in the UK; the 8-beam OCTAL laser¹⁵ at the Commissariat à l’énergie et aux énergies alternatives (CEA) at Limeil, France; the 9-beam Delfin laser¹⁶ at the Lebedev Physics Institute in the USSR; the 12-beam Orion laser¹⁷ at the Atomic Weapons Establishment, UK; and the 12-beam Iskra-5 laser¹⁸ at the Russian Federal Nuclear Center. (In this review, unless otherwise stated, OMEGA refers to the 60-beam laser.) Implosion facilities that are planned or under construction include the Laser Mégajoule¹⁹ near Bordeaux, France; the 128-beam Iskra-6 laser²⁰ at the Russian Federal Nuclear Center; and the SG-III (48-beam) and SG-IV lasers²¹ at the Research Center of Laser Fusion in China. Many significant advances have resulted from work at other laser facilities such as the Nike laser²² at the Naval Research Laboratory (NRL). The evolution of target-physics understanding has been very much an international effort; for example, results obtained on a relatively small laser facility at the Ecole Polytechnique in France²³ played an important part in the worldwide shift to shorter-wavelength lasers.

The outline of this review is as follows. Section II presents an overview of the main physics processes that are associated with direct-drive implosions. Section III describes the implosion process in more detail with reference to an ignition target design. This description is one dimensional (1-D), to illustrate how a target performs ideally. Later sections (Secs. XV and XVI) are concerned with two-dimensional (2-D) and 3-D effects, whose understanding is essential to target design.

Section IV reviews the early years of the laser fusion program, from the earliest concepts developed by Nuckolls²⁴ before the demonstration of the first laser until approximately 1980, when the field moved from infrared to short-wavelength lasers. The following sections are organized by topic: laser absorption (Sec. V), laser beam uniformity (Sec. VI), imprint (Sec. VII), implosion experiments (Sec. VIII), hydrodynamic stability (Sec. IX), coronal plasma physics (Sec. X), and thermal transport (Sec. XI). Implosion experiments include room-temperature implosions, used routinely to study most aspects of implosion physics, cryogenic implosions, which correspond more closely to ignition designs, and polar-drive implosions, developed for use on the NIF to study direct-drive physics on a system configured for indirect drive.

Section XII covers shock timing and equation-of-state studies motivated by the needs of ICF implosions. Section XIII reviews the evolution of cryogenic target systems and cryogenic target science from the early days when the need for the deuterium–tritium (DT) fuel to be in a solid (cryogenic) form was first recognized. Sections XIV (fast ignition) and XV (shock ignition) cover two alternative approaches to ignition. With a view toward the ultimate objective of inertial fusion energy (IFE), Sec. XVI reviews ignition and high-gain designs and Sec. XVII describes the evolution of concepts for IFE reactors.

While it is convenient to arrange much of the material by subject area, it must be recognized that there are strong interrelationships among the different areas, with many experiments depending on multiple physics processes, so that the categorization of some of the material can never be perfect. An attempt has been made to indicate many of these interrelationships through cross-referencing.

To provide a distinction between “mature” science, which is the focus of this review, and current developments, as well as for practical reasons, this review covers material published before October 2013.

Many texts and reviews on direct-drive ICF are available. Atzeni and Meyer-ter-Vehn²⁵ give a comprehensive treatment of ICF physics. An extensive review of the field prior to 1993 (in French) is contained in a three-part series of volumes edited by Dautray and Watteau.²⁶ Other texts include works by Duderstadt and Moses,²⁷ Kruer,²⁸ Yamanaka,²⁹ Velarde et al.,³⁰ Drake,³¹ and Pfalzner.³² Velarde and Carpintero-Santamaría²⁴ give an account of the history of ICF written by its pioneers. Early ICF work was reviewed in papers by Brueckner and Jorna,³³ McCall,³⁴ and McCrory and Soures.³⁵ More recently, Rosen³⁶ provided a useful tutorial and Lindl,³⁷ while focusing on indirect drive reviewed much of the physics that is common to both approaches. A recent review of direct-drive ICF is given by McCrory et al.³⁸ 

The critical physical processes relevant to direct-drive implosions are summarized in Fig. 2-1, which schematically shows the four main stages of a typical implosion. The target is represented as a layer of cryogenic DT ice inside a CH shell but could include a DT-filled foam layer. Typical ignition and high-gain targets have diameters from 3 to 5 mm and ice layer thicknesses from 160 to 600 μm (Secs. III A and XVI). The interior of the DT ice layer contains low-density DT gas in thermal equilibrium with the ice. At early times [Fig. 2-1(a)], laser light is absorbed by the target, leading to the ablation of target material (at the “ablation surface”) to form a hot plasma. The laser irradiation typically starts with a sequence of one to three low-intensity pulses, sometimes known as “pickets.” The corresponding pressure pulses generated in the plasma launch a sequence of shock waves that propagate into the target, compressing it. The laser pulse intensity then increases rapidly, typically to an intensity of ∼10¹⁵ W/cm², launching a strong shock that merges with the earlier shocks around the time that the shocks break out of the inner surface of the ice layer (Sec. III A). During this shock-transit stage, target modulations [imperfections in fabrication or modulations imprinted by laser-beam nonuniformities (Sec. VII)] evolve as a result of a Richtmyer–Meshkov (RM)-like instability (Sec. VII C). After the shock reaches the inner surface, a rarefaction wave moves outward toward the ablation surface and the target shell and ice layer (collectively known as the shell) begin to accelerate inward toward the target center. Important issues during this initial stage include the development of ablation-surface modulations as a result of laser imprint, the “feedout” of inner-surface modulations carried by the rarefaction wave to the ablation surface (Sec. VII D), the timing of the sequence of shocks to minimize shock preheating of the fuel (Sec. XII B), and laser–plasma interactions in the coronal plasma (Sec. X). Laser–plasma interactions can have undesirable effects including the production of energetic electrons (also known as “fast,” “hot,” or “suprathermal” electrons), leading to fuel preheat. X rays from the hot plasma surrounding the target can also lead to preheat.

The laser intensity increases during the acceleration phase [Fig. 2-1(b)]. Ablation-surface modulations grow exponentially because of the Rayleigh–Taylor (RT) instability (Sec. IX), while the main shock within the DT gas converges toward the target center. Exponential growth continues until the perturbation amplitude reaches ∼10% of the perturbation wavelength, when the instability growth becomes nonlinear. The greatest concern during the acceleration phase is the integrity of the shell. The ablation-surface modulations grow at a rate that depends in part on the shell adiabat α, defined as the electron pressure divided by the Fermi-degenerate pressure that the shell would have at absolute zero temperature. Larger adiabats result in thicker, lower-density imploding shells, larger ablation velocities, and better overall stability, but at the cost of lower overall performance. (The ablation velocity is the rate at which the ablation surface moves through the shell.)

The main shock wave moves ahead of the accelerating shell. Soon after it reflects from the target center and reaches the converging shell, the deceleration phase begins [Fig. 2-1(c)]. As the shell decelerates, its kinetic energy is converted into thermal energy and the DT fuel is compressed and heated. The attainable compression depends on the temperature of the DT at the start of the deceleration phase, and the maximum temperature depends on the kinetic energy of the shell. The greatest issue during the deceleration phase is the hydrodynamic instability of the inner surface of the shell. The deceleration instability is seeded by the feedthrough of ablation-surface modulations to the inner surface combined with the original inner-surface modulations. A major target design issue is the choice of α that optimizes performance by trading off stability with compressibility. Low values of α allow for high fuel compression, but at the cost of increased target instability.

Peak compression occurs in the final stage [Fig. 2-1(d)]. Fusion neutrons and x rays are produced together with charged particles (alpha particles for DT fuel) that, for NIF-scale targets, deposit their energy in the DT fuel, bringing more compressed fuel to fusion temperatures and leading to a propagating burn. This is commonly referred to as ignition.^2,3 In these targets, the first fusion reactions occur in a central “hot spot”—a high-temperature, low-density region surrounded by a lower-temperature, higher-density DT shell. The hot spot results from the compression of hot fuel in the deceleration phase and typically accounts for ∼10% of the compressed fuel mass. It is critical that the hot spot has sufficient energy production and areal density ρR (where ρ is the density and R is the radius) for significant alpha-particle energy deposition to occur. From Refs. 2 and 3, this typically requires an ion temperature of 10 keV and ρR ≳ 300 mg/cm², although an ion temperature of ∼5 keV at the onset of ignition with a larger ρR is considered to be more realistic. Another concern at this stage is mixing between the hot fuel in the core and the cooler shell material, reducing the temperature of the hot fuel.

One of the most important issues for direct drive is laser beam uniformity (Sec. VI). Long-wavelength nonuniformities, which arise from considerations such as the number of beams and their placement around the target chamber, can limit the target convergence. These nonuniformities are typically required to be less than ∼1% (rms). Shorter-wavelength nonuniformities, which are largely associated with the individual beam intensity profiles, are of concern as they can lead to laser imprinting (Sec. VII) during the initial stage of the laser–target interaction. Laser energy is deposited nonuniformly, according to the intensity distribution of each laser beam, resulting in ablation-surface modulations. Imprint occurs at early times until laser ablation produces a sufficiently large plasma in which the laser-absorption region becomes separated from the ablation surface. Thereafter, the shorter-wavelength laser-beam nonuniformities are smoothed by lateral thermal conduction (the “cloudy day” effect of Ref. 39) in the region between these surfaces. Imprint has been greatly reduced as a result of the development of laser beam-smoothing techniques (Sec. VI).

The properties of the hot coronal plasma surrounding the target are important for ensuring that the laser energy is efficiently absorbed and coupled to the imploding shell. The electron density n_e decreases with radius, passing through the so-called critical density n_c at the critical surface. At this surface, the electron plasma frequency, ω_p, defined by

where n_e, e, and m_e are the electron number density, charge, and mass, respectively, equals the incident laser frequency ω₀. The incoming laser light can propagate only up to the critical density. Since [from Eq. (2-1)] n_c is proportional to $ω02,$ a reduction in the laser wavelength by a factor of 3 (as for the frequency-tripled glass lasers currently being used for fusion research) increases the critical density by a factor of 9. (The critical density is given by $nc=1.115×1021/λ02 cm−3,$ where λ₀ is the laser wavelength in microns.) For frequency-tripled lasers, the main absorption process (inverse bremsstrahlung, Sec. V) involves electrons gaining directed energy through oscillations in the electromagnetic (EM) wave with this energy being thermalized through collisions. This process is most effective near the critical surface and favors short-wavelength lasers since the collision rate is larger at higher density. Energy is coupled more efficiently to the ablation surface of the imploding shell for short-wavelength lasers since it has a shorter distance to propagate; however, this shorter distance reduces the amount of thermal smoothing of laser nonuniformities, making laser beam smoothing particularly important. Energy transport from the absorption region to the ablation surface occurs through electron thermal conduction (Sec. XI), which is complicated by the existence of energetic electrons in the tail of the distribution function whose mean free paths between collisions are comparable to or larger than the temperature scale length.

A number of nonlinear laser–plasma interactions can take place in the portion of the plasma corona below the critical density (known as the “underdense” region) (Sec. X). Most notable are the two-plasmon–decay (TPD) instability, which occurs very close to the quarter-critical density and in which the incident EM wave excites two electron plasma waves (known as plasmons), and the stimulated Raman scattering (SRS) instability, which occurs below the quarter-critical density and in which the incident EM wave excites a scattered EM wave of lower frequency and a plasmon. These instabilities are of concern because electric fields associated with the plasmons can accelerate electrons to high energies. Another important process is stimulated Brillouin scattering (SBS), which is similar to SRS except that the plasmon is replaced by an ion-acoustic wave. This instability can occur anywhere below the critical density and can lead to a loss of energy. In one form of SBS, known as cross-beam energy transfer (CBET), laser energy can be lost by being scattered from incoming rays of one laser beam into outgoing rays of another. The effectiveness of these instabilities increases with laser intensity, imposing constraints on laser pulse-shape design.

Several nuclear reactions are of interest to laser-driven ICF. Most important are primary reactions from the DT fuel⁴⁰ 

The alpha particles (⁴He⁺⁺) are reabsorbed in the fuel of igniting targets, but the neutrons substantially escape with some energy loss that can provide a diagnostic of the fuel ρR. For energy production, the neutrons would be captured in the blanket of an inertial fusion reactor (Sec. XVII). Many experiments have used D₂ fuel, in which the primary reactions comprise two branches with approximately equal probabilities⁴⁰ 

The ³He and triton products from these reactions can combine with fuel deuterium in the following “secondary” reactions:⁴¹ 

where T^* and ³He^* indicate tritons and ³He nuclei that can have energies less than their corresponding birth energies [Eqs. (2-3) and (2-4)] because of slowing down in the compressed fuel. The primary reaction in D³He fuel,⁴⁰ 

is also important. This can be used to diagnose the mix by filling the target with ³He and placing deuterated plastic at some distance from the inner surface of a CH shell (Sec. VIII A).

To obtain hot-spot ignition of the DT fuel in an ICF implosion, a shell consisting of an inner cryogenic DT layer and an outer layer of ablator material is accelerated inward by a temporally shaped pressure drive. This drive is created by laser energy absorbed in the lower-density coronal plasma via inverse bremsstrahlung, at some distance from the higher-density main shell. The absorbed energy is transported by electrons and radiation to the shell, causing its outer layer to ablate and creating a force that accelerates the shell inward. As the shell approaches peak compression and stagnation, a hotter, lower-density central region (hot spot) surrounded by colder, higher-density main fuel is formed. The ion temperature and areal density of the hot spot must be high enough to create alpha particles (produced as a result of fusing D and T) at a rate sufficient that the hot-spot self-heating is larger than the hot-spot cooling (by radiation and hydrodynamic expansion), resulting in a burn wave being launched into the main fuel.

The laser-driven ICF process leading to hot-spot ignition is described in Sec. III A, with reference to a triple-picket ignition design for the NIF. (The term “picket” refers to a short laser pulse delivered prior to the main laser pulse.) The physics of this process is common to ignition and high-gain target designs. Section III B summarizes a simple model describing the ignition phase of the implosion and its scaling with laser energy. Section III focuses on 1-D physics processes. Multidimensional aspects of ignition and high-gain designs are included in Sec. XVI. Two alternative modes of ignition that do not use the main laser drive to form a hot spot are described in Sec. XIV (fast ignition) and Sec. XV (shock ignition).

The basic physics of ignition and high-gain target designs is illustrated using a 1.5-MJ triple-picket design for the NIF, similar to that shown in Ref. 42. (Here and elsewhere, the energy associated with a design refers to the incident laser energy.) The target and the laser pulse that drives it are shown in Fig. 3-1. The target is a 37-μm-thick plastic (CH) shell surrounding a 160-μm-thick layer of DT ice, with a DT vapor density of 0.6 mg/cm³ in the region inside the ice layer. The initial target radius is 1700 μm and the laser wavelength is 0.35 μm. The peak laser intensity is 8 × 10¹⁴ W/cm² relative to the initial target surface. The design assumes that the NIF is configured for spherically symmetric irradiation.

The required hot-spot pressure for a given drive energy can be achieved by shaping the laser pulse to increase the drive pressure and velocity of the main fuel as functions of time. Typically the drive pressure must reach ∼100 Mbar to achieve the compressed fuel pressure required for ignition. A steep rise of the drive pressure to ∼100 Mbar at the beginning of the pulse would lead to the formation of a very strong shock that would preheat the fuel, raise the shell entropy, and reduce the shell density resulting in a low final pressure. Therefore, the ignition target design strategy is to limit the strength of the initial shock launched into the fuel to a few Mbar to prevent a large increase in entropy. The first shock increases the fuel density by a factor of ∼4, allowing subsequent shocks to only marginally increase the entropy since the entropy jump across a strong shock is proportional to Δp/ρ^5/3, where Δp is the pressure jump and ρ is the pre-shock density. As the density increases because of multiple shock compressions, the entropy increase becomes smaller and the compression approaches an adiabatic compression. Using a sequence of judiciously timed multiple shocks, it is possible to keep the shell at a relatively low entropy while raising the drive pressure to ∼100 Mbar. This accelerates the shell to the required implosion velocity of ≥3 × 10⁷ cm/s.

The pressure increase from a few Mbar to ∼100 Mbar can alternatively be achieved adiabatically using a continuous-pulse design such as the “all-DT” design described in Sec. XVI. However, shock-velocity measurements in these designs⁴³ have shown that it is difficult to reproduce the adiabatic compression wave predicted by simulations. Because of this, current ignition designs are based on the generation of multiple shocks that can be accurately timed experimentally as described in Sec. XII.

In the ignition design of Ref. 42, the shocks are launched using a set of relatively short picket pulses followed by the main pulse. The picket pulses launch a sequence of decaying shocks that are experimentally timed to merge soon after they break out of the DT ice after propagating through the shell.

Precise shock timing is crucial to achieving high compression of the shell while it is in flight. A highly compressed shell acts like a rigid piston, efficiently transferring its kinetic energy to the central plasma contained within. Optimum shell compression is achieved when all shocks break out of the shell nearly simultaneously, minimizing the decompression resulting from rarefaction waves launched at each shock breakout. This requirement determines the picket amplitudes and timing. The use of multiple shocks has the additional benefit of reducing the shell nonuniformity growth factor associated with the Rayleigh–Taylor instability, as was pointed out by Lindl and Mead,⁴⁴ who presented the first multishock direct-drive design.

Figure 3-2 shows profiles of the mass density and pressure as functions of radius at a sequence of six times during the implosion from a simulation that excludes alpha-particle deposition to better illustrate the formation of the hot spot. In Fig. 3-2(a), just after the start of the main pulse, density and pressure discontinuities indicate the positions of the first three shocks and the main shock. An additional density discontinuity (contact discontinuity) occurs at the CH/DT interface.

As the shell accelerates inward, a shock wave propagates into the vapor region ahead of the shell, as seen in Fig. 3-2(b). The shock pressure in the vapor (∼1 Mbar) is much smaller than the ablation pressure (∼100 Mbar) at this time. The leading shock pressure increases later because of convergence. As the leading shock wave approaches the target center and reflects, the pressure produced in the central region is not yet large enough to slow down the converging main shell [Fig. 3-2(c)]. The main shell has a maximum implosion velocity of 4 × 10⁷ cm/s. The different stages of the deceleration phase and hot-spot formation are described in detail by Betti et al.^45,46 Continuous deceleration occurs when the vapor pressure has increased (because of convergence) to the level of the pressure of the main shell. As the shell continues to converge after that time, the vapor pressure keeps rising according to P_vapor∼ V^–5/3, where V is the volume of the vapor region, eventually exceeding the shell pressure and launching an outgoing shock (alternatively known as a return or rebound shock) through the incoming shell [Fig. 3-2(d)]. Before stagnation, the adiabatic approximation for the hot spot (P ∼ V^−5/3) is valid because the thermal conduction losses are recycled by ablation back into the hot spot. The central temperature increases during the compression and heat flows out of the hot spot, driving mass ablation off the inner shell surface. This causes the hot-spot mass and areal density to increase while keeping the hot-spot pressure unaffected by heat-conduction losses.⁴⁵ Bremsstrahlung radiation leads to hot-spot energy losses, but does not significantly affect the adiabatic scaling law since the radiation cooling rate of the hot spot is smaller than the rate of PdV work of the converging shell.

As the shell converges and the hot spot is compressed, the outgoing shock propagates through the incoming shell and slows down the imploding shocked material. Further deceleration occurs in the shocked region because of the pressure gradient pointing from the shock front toward the target center [Fig. 3-2(e)]. The imploding shell comprises two regions: shocked material and an outer region of incoming material. Figure 3-2(f) shows the density and pressure profiles at stagnation. As the shocked material stagnates, its kinetic energy is transferred into the internal energy of the hot spot and the shocked shell. The hot-spot pressure reaches its maximum value at peak compression. At this time, ∼1 mg of the initial fuel mass of ∼1.2 mg is in the shocked shell and ∼0.08 mg is in the hot spot.

The temporal behavior of the various shocks can be seen in Fig. 3-3, which shows contours of the inverse of the pressure scale length $|d(lnP)/dr|,$ where P is the total (electron plus ion) pressure. Since the pressure is discontinuous across a shock front, this plot is a convenient way of displaying shock trajectories from the simulation. (Numerically, the quantity plotted is never infinite because of the finite spatial resolution in the simulation.) The figure shows how the successive shocks are launched in response to sharp rises in the laser power and how the shocks are timed to reach the inner edge of the shell at around the same time, just prior to shell acceleration.

The hydrodynamic efficiency of this design, defined as the kinetic energy of the imploding shell divided by the absorbed laser energy, is 6.7%. The absorbed laser fraction is 95%. Not all the shell kinetic energy is converted into internal energy of the stagnated fuel. As seen in Fig. 3-2(f), the shock has not yet reached the outer boundary of the shell at the time of stagnation, so there is some fraction of the shell that is unshocked and imploding (in “free fall”). The kinetic energy of this part of the shell is not converted into internal energy before the inner surface stagnates and the hot spot reaches its peak pressure. Figure 3-4 shows the evolution of the internal energy of the hot spot (red curve) and the combined internal energy of the shocked material and the hot spot during deceleration as functions of time. Out of the 100 kJ of shell kinetic energy, 40 kJ is transferred into the internal energy of the hot spot at stagnation. Since the inner part of the shocked shell starts to expand outward by the time the outgoing shock wave reaches the outer boundary of the shell at 11.5 ns, the total internal energy (blue curve in Fig. 3-4) never reaches the value of the peak shell kinetic energy.

When the hot-spot areal density exceeds ∼0.2 g/cm², alpha particles start depositing a significant fraction of their energy inside the hot spot, raising its temperature and pressure. Once this self-heating is initiated, the adiabatic approximation is no longer valid. This is shown in Fig. 3-5, which gives profiles at 11.3 ns of mass density, ion temperature, and areal density, including alpha-particle deposition. A comparison of the temperature profile with the dashed curve (from a calculation with alpha deposition omitted) shows a rise in temperature resulting from alpha deposition. As the hot-spot temperature continues to rise and the number of alpha particles increases, the self-heating triggers a burn wave that propagates through the main fuel. The burnup fraction depends on the fuel areal density.^2,3,25 For this design, the target is predicted to ignite with a burnup fraction of 20% and produce a 1-D gain of 48.

Plots of electron density and temperature at the peak of the first picket (0.4 ns) and during the main pulse (8 ns) are shown in Fig. 3-6. The earlier density profile shows a small separation distance (standoff distance) D_ac= 10 μm between the critical surface (radius R_c) and the ablation front (radius R_a). This is a concern for laser imprinting (Sec. VII) because little lateral smoothing can occur over this small distance. The distance D_ac increases with time, reaching 170 μm at t = 8 ns, keeping the laser nonuniformities from further distorting the shell. The electron density scale length at quarter critical (2.25 × 10²¹ cm⁻³ for the 0.35-μm laser wavelength) is large (590 μm), leading to concerns about plasma instabilities producing hot electrons in this region (Sec. X).

Two important parameters of the 1-D design related to target stability, discussed further in Sec. XVI, are the in-flight aspect ratio (IFAR) and the convergence ratio. The IFAR (24.3 for this design) is defined near the beginning of shell acceleration, when the ablation-front radius is at 2/3 of the initial inner radius of the shell, as the ablation-front radius divided by the shell thickness. The convergence ratio (23 for this design) is defined as the initial inner radius of the shell divided by the inner shell radius at peak compression with alpha-particle deposition turned off.

Profiles of the in-flight shell density and the adiabat at t = 8.9 ns are shown in Fig. 3-7. The adiabat α is defined as the ratio of the shell pressure P_shell to the Fermi-degenerate pressure calculated at the shell density. It is a measure of the entropy added to the fuel by shocks and radiation. For DT fuel, the adiabat is given by

where the density ρ is in g/cm³ and P_shell is in Mbar. As shown in Fig. 3-7, the adiabat is “shaped” inside the shell. The larger value of α at the outside of the shell is favorable for hydrodynamic stability (Sec. IX). An ICF target design is usually characterized by the mass-averaged fuel adiabat, which is ∼1.6 for the design described here. The increase in α to the left of Fig. 3-7 results from the shock propagating in the DT gas shown in Fig. 3-2(b).

Adiabat shaping was suggested in 1991 by Gardner et al.⁴⁷ using controlled radiation deposition in the outer ablator, such that the ablation-front adiabat is raised (and the ablation-front density is lowered) without preheating the inner fuel layer. This radiative adiabat shaping has limitations, however, for typical ablator materials: the opacity is too low for DT, while the opacity of a solid-CH ablator is too high. Bodner et al.⁴⁸ pointed out that these problems could be avoided by using a CH-foam layer with DT wicked inside it as the ablator; the ablator would be covered with a thin high-Z layer, e.g., tungsten. Collins et al.⁴⁹ showed that the fluctuation decay lengths in the foam–fuel mixture are short enough that the shock jump (Rankine–Hugoniot) relations⁵⁰ are satisfied to within a few percent for shock strengths typical of ICF designs, validating the calculation of adiabats of shocked wetted foams with 1-D hydrodynamics codes. Goncharov et al.⁵¹ proposed an alternative approach, showing that tailoring the spatial shape of the adiabat can be accomplished by launching a decaying shock into the ablator. The pressure of this shock is higher in the ablator than in the fuel, thereby setting the ablator on a higher adiabat than the fuel. Later, Anderson and Betti⁵² found that a weaker decaying shock followed by a period of plasma relaxation and expansion would create a rarefaction density profile, which when shocked with a constant laser intensity yielded steeper adiabat gradients. Collins et al.⁵³ and Knauer et al.⁵⁴ confirmed this effect using 2-D simulations and experiments. Tabak independently investigated adiabat shaping using both the radiative and picket-pulse methods in the late 1990s.⁵⁵ Detailed calculations of both the decaying shock and relaxation methods of adiabat shaping were given by Anderson and Betti⁵⁶ and Betti et al.,⁵⁷ respectively.

The first picket provides the additional benefit of reducing laser imprint, as described in Sec. VII B 4.

Hot-spot self-heating is initiated if the rate of alpha heating exceeds the rate of hot-spot energy loss. This ignition condition, known as a Lawson-type criterion, can be written as⁵⁸ 

where P_hs is the hot-spot pressure in Mbar, τ is the hot-spot confinement time in seconds (the time required to disassemble the hot spot through hydrodynamic motion), ϵ_α= 3.5 MeV is the alpha-particle birth energy, and S(T_i) is a function of the ion temperature T_i at the center of the hot spot as given in Ref. 58.

The function S(T_i) is approximately linear with T_i up to a central temperature of ∼7 keV and reaches its maximum at ∼15 keV (see Figs. 4 and 5 in Ref. 58). It would be desirable to operate at the 15-keV maximum of S(T_i) to reduce the Pτ (product of pressure and confinement time) requirements for ignition. However, since the hot-spot temperature is approximately proportional to the implosion velocity, achieving ∼15 keV through compression would require such high implosion velocities that hydrodynamic instabilities would severely compromise the integrity of the imploding shell. Typical ICF ignition designs use a maximum implosion velocity in the range of 3.5–4.0 × 10⁷ cm/s, corresponding to an ion temperature of ∼4 to 5 keV.

Equation (3-2) can be rewritten by introducing an ignition parameter

Igniting the hot spot requires χ > 1. Assuming a perfectly spherical implosion and neglecting α-particle self-heating, Eq. (3-3) can be rewritten, using the hydrodynamic relations derived in Ref. 59, to relate χ to observables

where ρR is the total neutron-averaged areal density of the assembled fuel (including the hot spot and the main fuel) at peak compression in g/cm² and T_i is the neutron-averaged ion temperature of the hot spot in keV. Both areal density and ion temperature increase with the laser energy.⁵⁹ 

To assess the progress of direct-drive implosions using sub-ignition–scaled laser facilities, such as OMEGA, Eq. (3-4) must be rewritten in terms of hydrodynamic quantities that remain constant for subscale designs hydrodynamically equivalent to ignition designs on MJ-scale facilities. One of the most important parameters is the hot-spot pressure P_hs at peak compression. To write the ignition criterion in terms of this pressure, hydrodynamic scalings⁵⁹ are first used to rewrite Eq. (3-4) in terms of the hot-spot areal density and temperature

Since the DT fuel pressure (assuming equal ion and electron temperatures) is $PDT=2 ρTi/2.5 mp,$ where m_p is the proton mass, Eq. (3-5) takes the form

where R_hs is the hot-spot radius. Assuming that a fraction f_k of the total kinetic energy of the shell E_k is converted into the internal energy of the hot spot at peak compression, $fkEk=2πPhsRhs3,$ Eq. (3-6) can be rewritten in the form

Equation (3-7) shows that as the shell kinetic energy increases, the hot-spot pressure required for ignition decreases. For example, the target of Sec. III A couples ∼100 kJ out of 1.5-MJ incident laser energy into shell kinetic energy with f_k∼ 0.4 to 0.5. According to Eq. (3-7), this results in the minimum required hot-spot pressure exceeding 120–180 Gbar, which is smaller than the 215 Gbar predicted for this design. The maximum-allowed hot-spot size increases with the shell kinetic energy. This can be shown by combining Eqs. (3-6) and (3-7)

The condition on the minimum hot-spot pressure sets the requirements for laser pulse shaping and target dimensions. This can be understood using the following considerations: The hot-spot pressure increases by converting kinetic energy of the converging shell into internal energy of the stagnating fuel. Assuming that a fluid with velocity v and density ρ is stopped by a strong shock, the resulting pressure of the stagnated material is

This shows that the pressure at stagnation can be increased by increasing the shell density and velocity. For a given laser drive energy E_L, the shell velocity scales as⁴³ 

where P_a is the drive (ablation) pressure created by the ablated plasma blowing off the target, M_shell is the shell mass, and I is the intensity of the incident laser during the main portion of the pulse. Equation (3-10) shows that the shell velocity increases by reducing the shell mass and increasing the drive pressure at a given laser intensity (by using, for example, more-efficient ablator materials).

The scaling for the target radius R comes from an argument that the shell velocity scales with target radius R and acceleration time t_accel as v ∼ R/t_accel, where $taccel∼EL/R2I,$ so

Using Eq. (3-11) and writing M_shell∼ ρ₀Δ₀R², where ρ₀ is the initial average shell density and Δ₀ is the initial shell thickness, Eq. (3-10) takes the form

Equation (3-12) indicates that for a given ablator material (ρ₀ is fixed), the shell velocity can be increased by raising the laser intensity (which results in an increase in P_a) or by increasing the initial aspect ratio of the shell R/Δ₀. Several design limitations control the maximum values of both quantities. The maximum laser intensity is limited by the excitation of laser–plasma instabilities such as the two-plasmon–decay instability (Sec. X A) and stimulated Raman scattering (Sec. X C), which reduce the laser coupling and lead to the generation of suprathermal electrons that can preheat the main fuel layer. The maximum value of the shell aspect ratio is determined by multidimensional effects, in particular instability growth, which potentially can break up the shell if it is too thin.

The second contributing factor to the stagnating hot-spot pressure [see Eq. (3-9)], the shell density, is determined mainly by the fuel adiabat α. The fuel adiabat is controlled primarily by shock and radiation heating. In addition, the generation of suprathermal electrons from laser–plasma instabilities can increase the adiabat in some designs when the laser intensity exceeds the instability thresholds. Calculations⁴³ show that ignition can fail if 1% to 2% of the shell kinetic energy is deposited in the main fuel due to suprathermal electron preheat. The design shown in Fig. 3-1 reaches a kinetic energy of ∼100 kJ, leading to a limit of 1–2 kJ in preheat energy (or ∼0.1% of the incident laser energy) to avoid quenching the burn. Suprathermal electron preheat is discussed further in Sec. X A 2.

The laser-driven ICF concept originated in classified environments, with the key physics concept of compression predating the invention of the laser. A brief history given by Atzeni and Meyer-ter-Vehn in Sec. 3.4 of Ref. 25 includes descriptions of (1) the role of compression on fusion reaction rates, from work by Eddington on stellar energetics to imploding fission weapons and to controlled thermonuclear micro-explosions; (2) the proposals of several scientists (Nuckolls, Kidder, and Colgate in the U.S., and Basov, Krokhin, and Sakharov in the Soviet Union) immediately after the operation of the first lasers to use pulsed lasers to drive implosions; (3) the initiation of secret experimental programs in the 1960s; (4) a talk delivered by Basov in 1971 that led to the declassification of the compression concept and the publication of the seminal 1972 Nature article by Nuckolls et al.;¹ and (5) the later declassification of research on indirect drive.

A more-detailed history of ICF is given in Ref. 24—a collection of personal recollections written by pioneers of the field. The chapter by Nuckolls in Ref. 24 is particularly useful for its description of the origins of ICF and the now-declassified work in the period up to 1972. As recognized by Lindl,^2,3 the concept of laser-driven ICF grew out of work by Nuckolls in the late 1950s on the challenge of creating small fusion explosions without the use of an atomic bomb. Nuckolls postulated that a “non-nuclear primary” might be able to energize a radiation-driven implosion. He considered several candidates for the radiation source including plasma jets, pellet guns, and charged particle beams. (Later, he advocated heavy-ion accelerators for ICF power plants.) He calculated quantities such as the required mass of DT and the required radiation energy and temperature. The importance of isentropic compression to high densities, the associated need for a shaped drive pulse, and the problem of fluid instabilities were all recognized at this time. Some of Nuckolls' calculations were for the radiation-driven implosion of a “bare drop” of DT. After the demonstration of the laser, Lindl^2,3 credits Nuckolls and Colgate with calculating implosions in laser-driven hohlraums and Kidder with calculations that applied a spherically symmetric pulse of laser light directly to the target—the first direct-drive simulations.

Significant work was reported in the open literature in the 1960s that, while omitting the concept of compression, developed some important theoretical understanding and set in motion experiments aimed at demonstrating laser-induced fusion. Possibly the earliest publication on this topic is a report from the International Solid-State Circuits Conference in February 1961 (Ref. 60) in which Peter Franken discussed the heating of small pellets of lithium hydride to fusion temperature by a laser. Franken is quoted as warning that “the physics of this scheme do not favor success, but the work has, nevertheless, been started.”

In a 1962 conference abstract, Linlor⁶¹ presented what may be the first report of a laser-produced plasma and mentioned its possible application to controlled fusion research. Soon thereafter, in 1964, Dawson⁶² published ground-breaking theoretical work on laser-produced plasmas, covering many basic physics processes (such as inverse-bremsstrahlung absorption, thermal conduction, and electron–ion equipartition) that would subsequently be incorporated into hydrodynamic simulation codes. Dawson's main interest in this work was to use the laser to heat solid or liquid particles to fuel a magnetic-containment device. In what may be the first reference to the physical significance of the laser wavelength, he advocated higher-frequency lasers to heat the plasma at higher densities, improve the equipartition, and, thereby, heat the ions to higher temperatures.

In a later paper, Dawson et al.⁶³ considered the inverse-bremsstrahlung absorption process in more detail, including a finite density gradient in the plasma surrounding the solid particle. They recognized that efficient absorption is only possible with a sufficiently large scale length and estimated that a scale length of 1.4 mm would be needed to heat a deuterium plasma with a ruby laser (wavelength λ₀= 0.7 μm), assuming an electron temperature of 10 keV. Considering a 1.4-mm-radius sphere of electrons at the critical density, they estimated that an energy of the order of 100 kJ would be needed to accomplish this. (Of course, the density–radius product of this sphere, of the order of 1 mg/cm², would have been far too small for alpha-particle deposition.) They again advocated using higher-frequency lasers (“It might be advantageous to pass the light through a second harmonic generator”), noting that if the laser frequency is doubled, the density at the point where it is absorbed is 4× greater, allowing the radius of the particle to be reduced by a factor of 4.

While Dawson et al. did not take into account the need for compression and did not have appropriate hydrodynamic simulation codes available, their physical intuition that a deuterium plasma of a sufficient size, driven by a short-wavelength laser, could absorb a substantial fraction of the laser energy was correct and showed remarkable foresight. Their numerical estimates were mostly close, consistent with the parameters of the all-DT ignition design described in Sec. XVI, which has an outer radius of 1.7 mm and a coronal scale length of ∼0.5 mm, uses only inverse bremsstrahlung as an absorption mechanism, and is driven by a 0.35-μm laser. The only estimate that was a long way off was the laser energy (which is 1.5 MJ in the all-DT design); ignition designs account for the ∼10% conversion efficiency between the energy absorbed in the corona and the energy of imploding material and the energy cost associated with ensuring adequate target stability.

Unfortunately, the available laser energies, laser wavelengths, and target scale lengths in the 1960s and 1970s were such that the regime envisaged by Dawson et al. could not be realized. Instead, target physics was generally dominated by other absorption mechanisms and by their often undesired consequences.

On the experimental side, leadership was provided by the work of Basov and others at the Lebedev Institute in the USSR. In 1964, Basov and Krokhin^64,65 reported on using lasers to heat hydrogen to fusion temperatures. (Reference 65 referred to a paper delivered by Basov at the Presidium of the USSR Academy of Sciences as being one of the first suggestions to use lasers to initiate fusion.) In 1968, Basov et al.^66,67 reported the first observations of neutrons from solid LiD targets irradiated using short (∼10-ps), 1-μm-wavelength Nd:glass laser pulses up to 20 J. Their experimental configuration is shown in Fig. 4-1(a). Their large plastic scintillator counter allowed them to measure neutrons in a large solid angle. On some shots, a single neutron was detected, such as shown on the scope trace of Fig. 4-1(b). Basov et al. claimed that these neutrons were thermonuclear, but this proved to be controversial.

Similar measurements were soon reported by other laboratories, also using Nd:glass lasers. Gobeli et al.⁶⁸ of Sandia Laboratories confirmed the Lebedev experiments, also using LiD targets and with similar laser parameters (up to 25 J in short laser pulses). They found “a somewhat higher rate of neutron production,” meaning one to three neutrons per shot. Significantly, greater neutron counts were reported in 1970 by Floux et al.⁶⁹ at Limeil. They irradiated cryogenic D₂ targets using slightly larger energies (30–50 J) and substantially longer laser pulses (of ∼10-ns duration), at focused intensities of a few times 10¹³ W/cm². They observed 100–150 neutrons per shot. They too claimed a thermonuclear origin. They estimated the electron temperature to be between 500 and 700 eV, based on x-ray signals, and estimated that if ions were at a similar temperature, they would produce some fusion reactions. In 1971, Basov et al.⁷⁰ repeated the long-pulse experiments of Floux et al., except that they used CD₂ targets. They quoted 10³ neutrons per shot for a laser energy of 14 J.

In 1972, Mead et al.⁷¹ at Livermore reported 10³ to 10⁴ neutrons from solid CD₂ targets, but only from pulses longer than 2 ns or double pulses. The reproducibility was poor and not understood. At the Max Planck Institute in Garching, Büchl et al.⁷² observed up to 10³ neutrons from a solid D₂ target at low intensities (5 × 10¹² W/cm²) and found on the basis of time-of-flight measurements that the neutrons came from the target. However, based on their hydrodynamic predictions of just 10^–2 neutrons per shot and observations that the neutrons disappeared when a background gas was added, they concluded that the neutrons were not thermonuclear. An opposite interpretation came from further experiments at Limeil, in which Floux et al.⁷³ reported ∼10⁴ neutrons from solid D₂ targets heated by long pulses (∼3.5 ns) and higher intensities (∼8 × 10¹³ W/cm²). They claimed that their yield was thermonuclear, coming from the overdense region of the corona where the ion temperature T_i was predicted to reach ∼600 eV. They recognized that this explanation was consistent with the observations of lower yields for shorter pulses, where there was less time to heat the ions. Their claim must be considered to be highly plausible. Their results were supported by hydrodynamic calculations (incorporating the basic physics processes of inverse bremsstrahlung absorption, thermal conduction, and electron–ion equipartition described by Dawson⁶²) that predicted yields reasonably close to what was observed as well as a time-dependent reflected light fraction that was remarkably close to observations (Sec. X B). This explanation is supported by earlier theoretical work of Shearer and Barnes,⁷⁴ who had carried out some preliminary simulations of the early Lebedev experiments and identified the basic physics processes.

A multibeam irradiation geometry was reported by Basov et al.,⁷⁵ who used the nine-beam Delfin laser of Ref. 16 to focus ∼200 J onto a solid 110-μm-diam CD₂ target to obtain a yield of 3 × 10⁶ neutrons. They suggested that this geometry was favorable because it avoided the spreading of the heated region beyond the spot diameter that occurs for planar targets. This was not presented as an implosion experiment, and it is plausible that the neutrons again came from the overdense coronal region.

A thermonuclear source for the neutrons was not universally accepted. McCall et al.⁷⁶ at Los Alamos argued that the neutrons could be produced in the blowoff plasma by fast ions generated through electrostatic acceleration. They showed that neutrons could arise from deuterium monolayers deposited on the chamber wall. They obtained neutrons using a short-pulse laser (17 J, 25 ps) at a high focused intensity of ∼3 × 10¹⁶ W/cm², more likely to produce nonlinear interactions, and observed fast ions with velocities ∼2 × 10⁸ cm/s. Yamanaka et al.⁷⁷ at Osaka obtained up to 2 × 10⁴ neutrons from a solid D₂ target with 2-ns pulses at up to 10¹⁴ W/cm². This intensity is comparable to that of Floux et al.⁷³ and the pulse, although shorter, is long enough to establish a reasonable plasma scale length, so the plasma temperature must have been comparable and the similar observed neutron yield could also be consistent with a thermal origin. However, Yamanaka et al. observed fast ions (with an average speed of 10⁸ cm/s) and invoked them as the source of the neutrons. They also invoked the parametric decay instability (PDI) (Sec. X E) for much of the absorption.

In Ref. 78, Basov et al. agreed with McCall et al.⁷⁶ that in the case of tight focusing on a solid target, the neutrons were produced external to the plasma from accelerated deuterons. However, they found that in a multibeam irradiation geometry with spherical targets, the neutrons were thermal, on the basis of the width of the energy spectrum of the neutrons. In Ref. 79, Soures et al. reported over 10⁴ neutrons from a one-beam laser tightly focused onto solid spherical LiD targets; based on time-of-flight measurements, these neutrons originated from the target. This same conclusion was drawn by Basov et al.⁸⁰ from time-of-flight measurements in their multibeam system. Bodner et al.⁸¹ proposed that the early fusion neutrons originated from ions directly heated to keV temperatures in a narrow turbulent region near critical, resulting from the parametric decay instability. Boyer⁸² suggested that the SBS (Sec. X B) instability could have been responsible.

Spherical irradiation had been considered for some time. In 1966, Daiber et al.⁸³ proposed a laser-driven spherical implosion in which a first set of beams would be focused to a point in a gas, producing a spherical blast wave. A second set of beams would deposit their energy in the high-density shell of the blast wave, producing a spherically imploding shock wave. The intended application was fusion, and laser energies in excess of 10 MJ were envisaged for breakeven. Kidder⁸⁴ considered the implosion of a sphere of D₂ just above the critical density and found that a 100-kJ light pulse would produce 0.3 J of fusion energy. Mead⁸⁵ reported a spherically symmetric, 12-beam ruby laser system capable of 2 J per beam, in which 10-ns laser pulses were focused into a gas to produce spherical blast waves. Unfortunately, the energy had to be limited to 0.2 J per beam to avoid multiple breakdowns in the gas along the beam paths.

The field received a major impetus in 1972 with the Nature paper published by Nuckolls et al.¹ in which the concept of ablative compression to high densities was disclosed. This paper focused on solid DT spheres, although it was mentioned that shells could have some advantages as long as their aspect ratio was not too high. A central part of the concept, shown in Fig. 4-2(a), is a shaped laser pulse whose power increases rapidly in time over several orders of magnitude. (The peak power, 10¹⁵ W, exceeds that currently available on the National Ignition Facility by about a factor of 2.) The pulse shape was designed to isentropically compress the fuel to maximize the final density. Gain curves [Fig. 4-2(b)] were presented, showing that the required laser energy can be greatly reduced (e.g., to 1 kJ) if high compression can be obtained. Many important aspects of direct-drive physics were identified, as is evident from the following findings: (a) a sequence of about ten pulses should suffice to create the shaped pulse; (b) 20% uniformity could be produced with six overlapping beams, and this could be reduced to <1% by thermal conduction; (c) absorption occurs via inverse bremsstrahlung and plasma instabilities; and (d) there are problems with suprathermal electrons generated by laser–plasma instabilities preheating the fuel, these problems being worse for longer-wavelength lasers such as CO₂ (λ₀= 10.6 μm) and minimized for ultraviolet lasers. The concepts described in Ref. 1 were expanded upon in Ref. 86.

In a commentary written just after Ref. 1 that reflects the partial physics understanding of the time, Rosenbluth and Sagdeev⁸⁷ stated, “Under conditions of interest for laser fusion (electron temperatures of many keV) the collision frequency in hydrogen is such that [inverse bremsstrahlung] absorption will be negligibly small.” They added that, fortunately, the parametric decay instability “provides a very efficient mechanism for anomalous absorption.” This view was shared by Brueckner,⁸⁸ who claimed that “strong non-classical absorption occurs for laser power in excess of 10¹¹ to 10¹² W/cm²” and anticipated that the absorption efficiency would approach unity. Indeed, for many years thereafter, simulations by many computer codes would deposit up to 100% of the laser energy reaching the critical surface by assumed “anomalous mechanisms.” Rosenbluth and Sagdeev were also concerned about SBS and advocated reducing this by going to shorter-wavelength lasers to increase the collisional damping of the SBS ion-acoustic waves.

The next major step forward came with the demonstration at KMS Fusion of thermonuclear neutrons resulting from laser-induced compression, reported by Johnson et al.⁸⁹ and Charatis et al.⁹⁰ A schematic of this experiment is shown in Fig. 4-3(a). KMS used a unique ellipsoidal clamshell optical system⁹¹ to irradiate the spherical target with a high degree of uniformity. The targets were thin-walled glass microspheres, with diameter ∼70 μm and wall thickness ∼1 μm, filled with 13–18 atm of DT. Neutron yields were typically a few times 10⁵. Evidence for target compression was provided by Campbell et al.,⁹² who showed time-integrated x-ray pinhole camera images from empty glass microspheres (without the DT), such as the image in Fig. 4-3(b). The image shows emission from the outside of the target resulting from the hot coronal plasma and an intense central feature from the compressed core. Figure 4-3(c) compares a lineout of this image with three 1-D hydrodynamic simulations. The simulations incorporated angular momentum values L₁, L₂, and L₃ to represent deviations from spherical symmetry.

There was little doubt from the KMS experiments that the neutrons from microshell implosions were thermonuclear. This was confirmed by Slivinsky et al.,⁹³ who measured the ion temperature (∼3 keV) from the broadening of the alpha-particle time-of-flight signal. They also showed an energy downshift of 0.2 MeV from the original 3.52 MeV, demonstrating that the neutrons originated from inside the glass shell. The final confirmation that the neutrons were generated in the central compressed core was provided by Ceglio and Coleman,⁹⁴ who imaged the alpha particles obtained from two-beam implosions on the Argus laser at Livermore. Their reconstructed images showed concentric elliptical contours with an a/b ratio of 1.1–1.2 and a volume compression of ∼50. Ion-temperature measurements based on the neutron time-of-flight broadening (the neutrons were virtually unperturbed by passage through the compressed gas or glass shell) were reported by Lerche et al.⁹⁵ 

Energetic particles and x rays had been observed from laser-produced plasmas for a long time. Energetic x rays are understood to result from the bremsstrahlung radiation emitted by energetic (suprathermal) electrons as they slow down. Here, “energetic” must be understood in the context of the time; the actual energies that were found noteworthy tended to increase as more powerful lasers became available. Linlor⁹⁶ noted ion energies up to 1 keV using a ruby laser. Langer et al.⁹⁷ measured electrons, ions, and x rays emitted from plasmas produced by ruby and Nd:glass lasers (albeit at the low intensity of ∼6 × 10¹⁰ W/cm²) and suggested a space-charge–separation accelerating electric field. Using a Nd:glass laser, Büchl et al.⁷² observed a two-component x-ray spectrum with an ∼2-keV nonthermal component that disappeared (along with the neutrons) when a background gas was added. Shearer et al.⁹⁸ irradiated plastic targets with a Nd:glass laser at ∼2 × 10¹⁴ W/cm² and found 100-keV x rays with a temperature T_H∼ 50 keV; they suggested the parametric decay instability as the cause. Olsen et al.⁹⁹ reported 200- to 800-keV x rays, also from a Nd:glass laser. Ehler¹⁰⁰ observed proton energies reaching ∼100 keV at a CO₂ laser irradiance up to 10¹⁴ W/cm².

A typical x-ray spectrum is shown in Fig. 4-4(a) from experiments at NRL by Ripin et al.¹⁰¹ in which solid CH₂ targets were irradiated at ∼10¹⁶ W/cm² with 21-ps, 1-μm pulses. They claimed that magnetic fields, rather than a suprathermal electron component, were sufficient to explain the data (by inhibiting the thermal heat flux) and gave predictions of a magnetic-field model that came close to the data. Interestingly, the “thermal” temperature predicted by their model, ∼60 keV, was higher than the maximum typically observed suprathermal temperature of ∼20 keV.

The 60-keV temperature may have arisen from an assumption that all of the incident laser light was completely absorbed when it reached the critical density. This procedure, commonly referred to as a “dump-all,” was routinely used in simulations at the time such as those of Refs. 102–105. In Ref. 103, all the energy was dumped in the first overdense zone, generating a sharp temperature gradient even though the mean free path of a 20-keV electron at n_e= 10¹⁹ cm⁻³ is ∼100 cm. Other simulation models allowed more flexibility; e.g., in the model described by Mead et al.,¹⁰⁶ the user was allowed to specify the fraction of a ray energy that was dumped and the density that the ray needed to reach for this to happen.

Another x-ray spectrum, obtained from CD₂ targets irradiated at 1-μm wavelength at the much lower intensity of ∼2 × 10¹³ W/cm² by Slivinsky et al.,¹⁰⁷ is shown in Fig. 4-4(b). While the x-ray signal fell off faster with photon energy as a result of the lower irradiance, it exhibited similar behavior in that it could not be modeled using just classical inverse-bremsstrahlung absorption. Slivinsky et al. explained their results with “anomalous absorption,” but they were unable to tell whether this was resonance absorption or a parametric instability. A third explanation was offered for similar results (Fig. 11-2 below) obtained in earlier work by Kephart et al.¹⁰⁸ in terms of thermal flux inhibition (Sec. XI).

A consensus emerged that the hard x rays resulted from suprathermal electrons produced by resonance absorption. In this absorption process, described in more detail in Sec. IV C, p-polarized light incident with a non-normal angle of incidence has a turning point below critical but tunnels to the critical density where it resonantly excites electron plasma waves, which accelerate electrons to high energies. Typically the hard x-ray spectrum showed an exponential falloff with x-ray energy, characterized by a hot temperature T_H. (This is difficult to see in Fig. 4-4 because of the logarithmic x axes.) Brueckner^109,110 gave a convenient way of estimating the total hot-electron energy from the hard component in the spectrum.

A compilation of results by Lindman¹¹¹ for the T_H obtained from a large number of laboratories is shown in Fig. 4-5(a) for Nd:glass lasers (λ₀= 1.06 μm) and CO₂ lasers (λ₀= 10.6 μm). Remarkably, when plotted against $ILλ02,$ where I_L is the laser intensity, all data lie very close to the same curve [Fig. 4-5(b)]. This curve and its interpretation as resonance absorption are consistent with particle-in-cell (PIC) simulations carried out by Forslund et al.^112,113 and by Estabrook and Kruer.¹¹⁴ The parameter $ILλ02$ is important since it is proportional to the oscillatory energy of an electron in the electromagnetic field. Different scalings were noted for two regimes, described as “weak” and “strong” profile modification. Profile modification refers to the steepening of the density profile near critical. This was commonly ascribed to radiation pressure, sometimes described as the ponderomotive force, although there were alternative explanations (Sec. IV D). Early work on the ponderomotive force is described by Hora.¹¹⁵ PIC simulations by Estabrook et al.¹¹⁶ indicated that resonance absorption caused the steepening and was also enhanced by the steepening.

While scaling laws can be very informative, an understanding of their regimes of applicability is equally important. If one were to extrapolate the $ILλ02$ scaling of Fig. 4-5 to the λ₀= 0.35-μm ignition design of Sec. III at 10¹⁵ W/cm², one would find T_H just below 2 keV, lower than the expected thermal temperature of ∼4 keV, in contradiction to the premise that T_H represents electrons hotter than thermal.

Very informative x-ray images of spherical glass microshells (diameter ∼300 μm, thickness ∼1.5 μm) imploded on the 1-μm-wavelength Shiva laser were obtained by Ceglio and Larsen¹¹⁷ using zone-plate coded imaging. The laser delivered 17–20 TW in 90-ps pulses. The Shiva beams were arranged in two clusters of ten beams on either side of the target (in reality, above and below the target) with incidence angles up to 17.7°, so the target experienced strongly two-sided irradiation. Two (time-integrated) images are shown in Fig. 4-6: for thermal x rays [Fig. 4-6(a)] and for suprathermal x rays [Fig. 4-6(b)]. The thermal x rays come primarily from the hot compressed core, the two-lobed structure being associated with the two-sided irradiation. The suprathermal x rays come primarily from the outer target surface, with maximum emission from the portions of the target directly facing the laser beams.

From Fig. 4-6(b), it is evident that hard x rays also come from the less-irradiated region of the target in the center of the figure (the laser axis is horizontal in the figure, as indicated by the arrows). This is consistent with a picture in which the hot electrons can pass through the shell, fuel, and corona multiple times before slowing down. The electrons can lose their energy in two ways: (1) as a result of collisions in the shell, leading to x-ray emission and (2) as a result of reflections from a moving sheath on the outside of the corona, leading to the acceleration of fast ions. In this picture, the electrons provided considerable thermal smoothing of the deposited energy, enabling reasonably symmetric implosions of targets that were often irradiated with only two beams (or two clusters of beams as on Shiva).

These targets are known as “exploding pushers” because the glass shells, heated throughout their volume by the hot electrons, exploded. Exploding-pusher targets were typically irradiated at high laser intensities (to generate high temperatures) and were useful for producing neutrons. However, it was recognized that they would not scale to breakeven and gain because the large fuel preheat would preclude high compression. The short pulses used to drive these targets bore no resemblance to the shaped pulses required by Nuckolls et al.¹ for ablative compression. Storm¹¹⁸ developed a model for exploding-pusher targets that enabled the dependence of yield on parameters such as laser pulse width, target diameter, wall thickness, and fill pressure, to be predicted without recourse to more-accurate hydrodynamic simulations that, at this time, required significant computational time even in one dimension. The model scaled with experiments and calculations to within a factor of 2 over several decades of neutron yield. Giovanielli and Cranfill,¹¹⁹ Rosen and Nuckolls,¹²⁰ and Ahlborn and Key¹²¹ also developed exploding-pusher models.

While the parametric decay instability had in earlier years appeared to be the primary laser-absorption mechanism, the success of the T_H scaling coupled with PIC simulations led to resonance absorption becoming the favored mechanism. The steepening of the density profile near critical worked against the parametric decay instability but helped resonance absorption because the incident wave had to tunnel through a shorter distance to reach critical. This was formalized by the basic resonance absorption function ϕ(τ) [Fig. 4-7(a)], published by Denisov¹²² in 1957 in the context of the propagation of electromagnetic radiation in the ionosphere and later by Ginzburg.¹²³ The quantity τ is defined as τ = (2πL/λ₀)^1/3sin θ, where L is the scale length at critical and θ is the angle of incidence, and ϕ(τ) represents the amplitude of the tunneled electric field. A similar curve is given by Friedberg et al.¹²⁴ for the absorption in cold plasmas. They found a maximum absorption of ∼40% at θ = 23° for λ₀= 1.06 μm and a steep density profile with scale length L = 1.7 μm. They noted that the energy of the resonantly excited electron plasma waves could be transferred to energetic electrons by means of wave breaking and Landau damping. The work of Ref. 124 was extended to hot plasmas by Forslund et al.¹¹² 

The theory of resonance absorption made several clear predictions: no absorption at normal incidence, no absorption for s polarization, and a strong angular dependence for p polarization. The results of one of the very limited number of experiments that set out to investigate these predictions are shown in Fig. 4-7(b), from Manes et al.¹²⁵ and Thomson et al.¹²⁶ They used a box calorimeter surrounding the target together with calorimetry of the light backscattered through the focus lens to measure the total scattered light, thereby obtaining an accurate measurement of the absorption. They irradiated CH disk targets with ∼30-ps, 1-μm-wavelength pulses at ∼10¹⁶ W/cm² with a small focal diameter of ∼40 μm. They indeed observed a difference between the two polarizations in the angular dependence, indicating resonance absorption. However, the observed angular dependence for p polarization bore little resemblance to the curve of ϕ(τ). The absorption fractions observed at normal incidence and for s polarization were much larger than would have been expected for inverse bremsstrahlung and were more difficult to explain. Thomson et al. offered the explanation that the critical surface was rippled, as postulated by Estabrook et al.¹¹⁶ (More broadly, the whole density profile in the vicinity of critical could be considered to be strongly inhomogeneous.) However, the PIC simulations of Ref. 116 predicted ∼15% absorption at normal incidence caused by the parametric decay and oscillating two-stream instabilities (Sec. X E), so the interpretation of Fig. 4-7(b) remains inconclusive.

Figure 4-7(c) shows data from an experiment at Garching by Godwin et al.,¹²⁷ who used an Ulbricht spherical photometer [similar to that shown in Fig. 5-1(b) below] to measure the scattered light. They irradiated Cu targets with 400-mJ, 30-ps pulses at a 1-μm wavelength over a wide range of intensities up to 10¹⁶ W/cm² and plotted the reflectance. Their results were similar to those of Fig. 4-7(b): there was about 30% absorption at normal incidence and the difference between s and p polarization was slightly larger. Similar experiments were reported for 0.5-μm irradiation by Maaswinkel et al.¹²⁸ 

The plasma scale length L is of interest because it appears in the resonance absorption function ϕ(τ) and, more broadly, because it affects the amount of inverse bremsstrahlung and the thresholds of parametric instabilities. An early measurement of the time-dependent scale length was provided by Jackel et al.¹²⁹ using the four-beam DELTA laser. By streaking the 2ω₀ and $3ω0/2$ emissions from ∼40-μm-radius glass microballoon targets [Fig. 4-8(a)], where ω₀ is the incident laser frequency, they were able to plot the critical and quarter-critical surfaces as functions of time [Fig. 4-8(b)] and measure separation distances up to ∼40 μm. [Second-harmonic emission, observed by Decroisette et al.¹³⁰ in 1971, is understood to result from nonlinear processes at the critical surface, and the $3ω0/2$ emission, a signature of the two-plasmon–decay instability (Sec. X A), comes from quarter critical.] Time-integrated images of the 2ω₀ and $3ω0/2$ emissions were first reported by Saleres et al.¹³¹ for solid D₂ targets.

An interferometric measurement of the density profile was obtained by Attwood et al.¹³² for the one-sided irradiation of 40-μm-diam glass microballoons with 30-ps pulses at ∼3 × 10¹⁴ W/cm². A typical interferogram, obtained using a 15-ps, 0.266-μm, frequency-quadrupled probe, is shown in Fig. 4-9(a), with the laser incident from the right. Using Abel inversion in different planes perpendicular to the laser axis, they produced the reconstructed density profile of Fig. 4-9(b). This showed the predicted steepened profile between critical and 0.3× critical, together with upper and lower density “shelves.” Attwood et al. interpreted the steepened profile as confirmation of radiation-pressure effects, although Estabrook et al.¹¹⁶ maintained that heating had a stronger effect on the profile. Also using interferometry, Fedosejevs et al.¹³³ observed steepening in CO₂ plasmas and claimed direct evidence of radiation pressure, but they recognized that momentum deposition from particles accelerated away from the critical region to lower densities could be a factor.

Attwood et al.¹³² chose small targets to minimize refractive effects; however, the Abel inversion procedure neglects refractive effects. This was later investigated by Brown,¹³⁴ who analyzed the interferogram of Fig. 4-9(a) including refraction. He propagated rays through the reconstructed profile of Attwood et al. and found that no ray that was collected in the optical system reached a density higher than 7 × 10²⁰ cm⁻³ and that the reconstructed profile could not have produced the interferogram. Brown then found another three-slope profile, shown in red in Fig. 4-9(b), which was consistent with the interferogram. The upper shelf started at a lower density, and the maximum density reached by the probe was 1.0 × 10²¹ cm⁻³. Interestingly, he was unable to find any two-slope profile that matched the data. The more-accurate reconstruction confirmed the main observation of profile steepening but also made it clear that the role of refraction can be significant for the optical diagnosis of plasmas. Refraction would also have affected the observations of Jackel et al.¹²⁹ 

In the context of the rippled-critical-surface scenario proposed for the resonance absorption data of Fig. 4-7, one might wonder whether the clear, smooth fringes of Fig. 4-9(a) are consistent with anything but a minimum of rippling. In addition, the experiments of Fig. 4-7 were both for very short laser pulses that are unrepresentative of most laser–plasma interaction experiments. While the consensus based on the x-ray spectra is that resonance absorption was dominant for both long- and short-pulse irradiation at high intensities with infrared lasers, the direct evidence for resonance absorption is minimal. One would need to create a 3-D model of a rippled critical surface and then pass a probe beam through, including refraction, to see the effect on the interferogram.

Although far removed from ignition conditions, implosions driven by infrared lasers stimulated the development of new diagnostics, some of which were reviewed by Attwood.¹³⁵ One such diagnostic was described by Attwood et al.,¹³⁶ who coupled an x-ray pinhole camera to an x-ray streak camera to diagnose an implosion. They obtained spatial and temporal resolutions of 6 μm and 15 ps, respectively. Sample output is shown in Fig. 4-10. It became possible to follow the implosion continuously, through the compression and disassembly phases, based on the self-emission of the target. Streaked x-ray imaging of imploding targets was also reported by Billon et al.¹³⁷ 

Another advance, with far-reaching significance, was the development of x-ray backlighting at the Rutherford Laboratory using the Vulcan laser. This was reported by Key et al.,¹³⁸ who used the configuration shown in Fig. 4-11(a). A microballoon was imploded using two beams and radiographed by x rays produced from a brass (Cu + Zn) target, irradiated with a delayed laser pulse. The key feature of x-ray backlighting is that it makes it possible to diagnose cold dense plasmas such as imploding shells. A series of images obtained from a number of implosions with different backlighter timings are shown in Fig. 4-11(b), following the implosion through compression. This was accomplished with just 5 J of backlighter energy. Today, tens of kilojoules of backlighter energy are available on the NIF.

Key et al.¹³⁹ reported the first demonstration of time-resolved x-ray backlighting, in which a slit across the image of an x-ray microscope was coupled to a streak camera with 80-ps temporal resolution. Results for two imploding microballoons of different aspect ratio (radius divided by thickness) are shown in Fig. 4-12. The shell (a) with a small aspect ratio of 12 imploded to a radius that was close to the predicted minimum diameter, while the shell (b), with an aspect ratio of 110, appeared to have broken up, presumably because of the Rayleigh–Taylor instability.

Many diagnostics were developed at LLNL in the 1970s beyond those described in Ref. 135. An extensive review of these diagnostics and the experiments that used them is given by Ahlstrom.¹⁴⁰ 

Perhaps the most-significant advance in this period of time was the unequivocal demonstration at École Polytechnique by Fabre et al.²³ of improved absorption with short-wavelength lasers. With just 2.8 J of energy at the fourth harmonic of a Nd:glass laser, they obtained ∼90% absorption in the range of 10¹⁴–10¹⁵ W/cm². After years of experiments in which the absorption was dominated by mechanisms such as resonance absorption and parametric instabilities that were, at best, partially understood, it was refreshing to enter a regime, anticipated by Dawson,⁶² in which the absorption was dominated by inverse bremsstrahlung. The benefits of short-wavelength lasers were also important for indirect drive.^2,3 The experiments of Fabre et al. along with others at different laboratories under a variety of target conditions are described in Sec. V.

The results of Fabre et al.,²³ reported at a conference in 1979, stimulated a change in focus at LLE from the second harmonic of the Nd:glass laser to the third harmonic. A few months later, Seka et al.¹⁴¹ demonstrated third-harmonic generation with an efficiency of 80% using a tripling technique proposed by Craxton¹⁴² that was subsequently implemented on the 24-beam OMEGA laser at LLE, the Nova laser at LLNL, the 60-beam OMEGA laser, and other laser systems. The 192-beam NIF used a different technique, also proposed in Ref. 142. The impact of a single 2.8-J UV beam was enormous. Conversion of large Nd:glass lasers to the fourth harmonic was not pursued because of the lack of suitable crystals to perform the frequency conversion at high fluence and low fourth-harmonic damage thresholds.

The problems encountered in the understanding of laser–plasma interactions at 1 μm did not impede progress in the production of larger numbers of neutrons as more powerful lasers were built. McCrory and Soures³⁵ compiled Fig. 4-13 from exploding-pusher experiments carried out at LLE, LLNL, and Osaka. It is notable that the highest yields were obtained for short laser wavelengths. The solid blue circle showing 10¹⁴ neutrons and a thermonuclear burn energy of ∼1% of the laser energy, added subsequently to Ref. 35, is from the 60-beam OMEGA laser after its completion in 1995 (Ref. 143). Exploding-pusher implosions have received little attention since, except as convenient sources of neutrons and other fusion products for various applications. After the work of Fabre et al.,²³ ICF experiments became focused on pursuing the ablative type of implosion envisaged by Nuckolls et al.¹ in 1972.

The observations by Fabre et al.²³ of dramatically increased absorption at short laser wavelengths (understood in this section to mean wavelengths ≤0.53 μm) stimulated several similar experiments at other laboratories in the early 1980s. The main conclusion of Fabre et al. was confirmed, and it became clear that the dominant absorption mechanism at short laser wavelengths, in the intensity regime of interest for ICF, is inverse bremsstrahlung. The observations of these experiments are reviewed in this section, together with later 0.35-μm experiments on OMEGA in spherical geometry.

Although it is basic to the understanding of laser-produced plasmas, the fraction of the incident laser energy that is absorbed in the target is very difficult to measure accurately. Basov et al.¹⁶ recognized this problem and measured the absorbed energy by observing the time evolution of the expanding shock (blast) wave in the residual gas in their target chamber. Other workers relied on the light returning from the target through the focus lens (which was easy to measure), assuming that whatever light did not return into the focus lens was absorbed; however, this generally provides estimates of absorption of unknown accuracy since there is no measurement of light returning at other angles. The least-reliable estimates of the absorption are likely to be obtained for lenses of large f number (i.e., small solid angle), as discussed in Sec. X B with reference to Fig. 10-11.

The key to the accurate absorption measurements that were made on planar targets in the years around 1980 was the development of 4π calorimeters. Two such devices are shown in Fig. 5-1. Figure 5-1(a) shows a box calorimeter from Ref. 144, originally developed at LLNL by Gunn and Rupert¹⁴⁵ to measure the total unabsorbed light from experiments such as those of Fig. 4-7(b). The analysis of the calorimeter signals by Seka et al.¹⁴⁴ allowed the total plasma energy (ions plus x rays) to be measured, as well as the scattered light. Figure 5-1(b) shows an Ulbricht sphere, as used by Fabre et al.²³ The Ulbricht sphere was developed by Godwin et al.¹²⁷ at Garching and used for the measurements of Fig. 4-7(c). The inside of the sphere is coated with a scattering paint, allowing each photodiode to sample light scattered from the target into most of 4π steradians. For both devices, light returning into the focus lens is measured separately. Calibration of the Ulbricht sphere is described in Ref. 127.

The results from Ref. 23 were published by Garban-Labaune et al.¹⁴⁶ and are shown in Fig. 5-2 for a variety of laser wavelengths and for short and long pulses, all focused onto solid planar targets. Most data are for C₁₀H₈O₄, but the 0.26-μm-wavelength data include Al and Au. The increase in absorption as the laser wavelength decreases, in the intensity range (≲10¹⁵ W/cm²) of interest to direct-drive ICF, is dramatic. Theoretical modeling of the data was reported by Garban-Labaune et al.¹⁴⁷ 

Results from four other absorption experiments, carried out at three different laboratories in planar and spherical geometries and at 0.53 μm and 0.35 μm, are shown in Fig. 5-3. The results include 0.53-μm data obtained by Mead et al.¹⁴⁸ at LLNL and Slater et al.¹⁴⁹ at KMS Fusion, and 0.35-μm data obtained at LLE by Seka et al.¹⁴⁴ and Richardson et al.¹⁵⁰ Most datasets cover a large range of incident intensity, typically two to three orders of magnitude, by a combination of varying the laser energy and the focal-spot size.

The data in Fig. 5-3(a) were obtained from the irradiation of flat targets with one beam of the Argus laser in 600-ps pulses,¹⁴⁸ using a box calorimeter to measure the scattered light. Slightly higher absorption was found for Au compared with lower-Z targets (CH, Be), and in both cases the results were in agreement with hydrodynamics simulations using the code LASNEX. The low-Z results were also consistent with simulations using the hydrodynamics code SAGE carried out by Craxton and McCrory.¹⁵¹ Mead et al.¹⁴⁸ found low levels of hot electrons—less than ∼1% of the incident energy was converted into hot electrons for all target Z's at ∼2 × 10¹⁵ W/cm².

The KMS experiments [Fig. 5-3(b)] were carried out in spherical geometry, using the ellipsoidal-mirror target irradiation system⁹¹ shown in Fig. 4-3(a). The absorbed energy was determined with a differential plasma calorimeter at one location on the target chamber, so its accuracy depended on the assumption that the plasma blowoff was isotropic. Given the near-4π irradiation solid angle, this assumption is probably reasonable, although it would be hard to quantify. In addition to the 0.53-μm-wavelength data, which match the SAGE predictions, Fig. 5-3(b) shows data for 1 μm with the weaker dependence on intensity that was typically seen for experiments with a large (presumed) resonance absorption component.

The first measurements of absorption at 0.35 μm at LLE were reported by Seka et al.¹⁴⁴ and are shown in Fig. 5-3(c) together with SAGE simulations. They were carried out on the one-beam Glass Development Laser (GDL) System with UV energies up to 22 J at 90 ps and 60 J at 450 ps, using the box calorimeter of Fig. 5-1(a). The intensity was varied over two orders of magnitude by changing the spot diameter (100–800 μm) and the laser pulse energy. The absorption was generally found to be independent of spot size: no dependence on spot size was observed below 5 × 10¹⁴ W/cm², and small (∼5%) differences between 1-D planar simulations and 2-D cylindrically symmetric simulations were noted at high intensities caused by spherical divergence effects from small focal spots. The simulations all used a flux limiter (Sec. XI) of f = 0.03 to model electron heat transport and, in addition to inverse bremsstrahlung, deposited an ad hoc fraction of 15% (representing additional absorption mechanisms such as resonance absorption) of whatever light reached critical. All the SAGE simulations shown in Fig. 5-3 used these same parameters, demonstrating consistency across the different experimental configurations. Almost identical predictions were obtained for the four data sets of Fig. 5-3(c) using f = 0.04 and no ad hoc deposition at critical.¹⁴⁴ 

The data of Fig. 5-3(d), reported by Richardson et al.,¹⁵⁰ are from the original (24-beam) OMEGA Laser System after the conversion of six beams to the third harmonic. Solid-CH spherical targets of diameters between 50 and 400 μm were irradiated with 600- to 700-ps pulses in the intensity range 10¹³–10¹⁵ W/cm². For optimal uniformity, the best focus of the beams was placed beyond the target center so that the edge rays of the beam were tangentially incident on the edge of the target. The absorption was measured by a set of 20 differential plasma calorimeters situated symmetrically about the target. With this large number of detectors, the overall absorption measurement should have been minimally affected by irradiation nonuniformities. The large dynamic range in intensity was possible because phase plates (Sec. VI A) were not implemented on the laser beams at the time. There were significant hot spots in the laser beams [see Fig. 6-11(a) below], but their effect on the overall absorption was probably minimal since a combination of thermal transport at densities below critical and refraction through the plasma of the obliquely incident laser rays could be expected to ensure a fairly uniform electron-temperature distribution in the corona. The data were consistent with the predictions of the hydrodynamics codes LILAC and SAGE, which used the same flux limiter of 0.03 and a 15% deposition factor at critical.

Overall, the results shown in Figs. 5-2 and 5-3 from different laboratories, under different experimental conditions and across a wide range of incident laser intensities, demonstrated conclusively that inverse bremsstrahlung is the dominant absorption mechanism in short-wavelength experiments.

Further confirmation of the inverse-bremsstrahlung process was provided by Mead et al.,¹⁵² who measured the absorption of Au disk targets as a function of the angle of incidence (θ) using the Argus laser at 0.53 μm. Figure 5-4 shows results for an incident intensity of 3 × 10¹⁴ W/cm². The dependence of the absorption A on θ matches closely the prediction A(θ) = 1 − exp(−α₀ cos³θ) for inverse bremsstrahlung in an isothermal plasma with an exponential density gradient, where α₀ gives the absorption at normal incidence. (In contrast, Drake et al.¹⁵³ found no resolvable dependence of absorption on θ for all angles up to 50° for Au targets irradiated at 0.35 μm on one beam of the Nova laser; the reasons for this were not understood.)

Absorption on the 60-beam OMEGA laser has been routinely measured using scattered-light calorimeters placed at different angles around the target chamber. Two of these calorimeters are located in full-aperture backscatter stations (FABS), known as FABS25 and FABS30, which sample the light returning along the paths of Beamlines 25 and 30, respectively. The measurements in these calorimeters are adjusted to account for a small amount of “blow-by” light from the opposite beams, which is not intercepted by the target or the plasma. The accuracy of these measurements depends on the extent to which the scattered light is isotropic. While this can depend on the irradiation conditions, isotropy is generally a reasonable assumption. Examples of the absorption measured by these two backscatter stations¹⁵⁴ are shown for room-temperature implosions in Fig. 5-5(a) and for cryogenic implosions in Fig. 5-5(b). Each dataset spans a period of several months and includes data from a variety of laser pulse shapes with the peak laser intensity varying from 3 × 10¹⁴ to 1.1 × 10¹⁵ W/cm². The absorption fractions measured using the two different stations track each other closely as well as predictions of the code LILAC. While the pulse shape and target characteristics influence the absorption, the laser intensity was found to be the dominant factor with the higher absorption fractions being associated with the lower intensities.¹⁵⁴ 

Streaked spectrometers on OMEGA have provided further insight into the absorption process. A typical streaked spectrum, obtained by Seka et al.,¹⁵⁵ is shown in Fig. 5-6(a) for an imploding cryogenic target irradiated by a single-picket pulse, observed in a port between beams that is not subject to blow-by effects. In this case, 1-THz, 2-D smoothing by spectral dispersion (SSD) (Sec. VI C) and polarization smoothing (Sec. VI D) were used with a ∼0.4-nm UV bandwidth. The initial blue shift comes from the reduction in optical path for rays passing through the expanding plasma,¹⁵⁶ and the later red shift is associated with opposite changes to the optical path as the target implodes. (The refractive index $μ=1−ne/nc,$ where n_e is the electron density and n_c is the critical density; the optical path ∫μds, where s is the distance along a ray path, therefore decreases when n_e increases along the path.) By integrating over the spectrum, the scattered-light power is obtained as a function of time [Fig. 5-6(b)]. This is shown together with the incident laser pulse and compared with the LILAC simulations using a flux-limited model with f = 0.06 and a nonlocal heat-transport model.¹⁵⁷ The differences between the observations and the predictions are typical for these experiments, but the details depend on target and irradiation parameters.

Seka et al.¹⁵⁵ made similar measurements for experiments using 200-ps pulses to investigate the physics associated with absorption in the picket portion (see Fig. 3-1) of implosion pulse shapes. Here, the nonlocal model provided a much better match to the experimental data (Fig. 5-7). Some work that advocates using a higher flux limiter at early times (which would decrease the predicted scattered light of the flux-limited curve in Fig. 5-7) is described in Sec. XI with reference to Fig. 11-8.

In more recent work, Igumenshchev et al.¹⁵⁸ have reported experiments in which the scattered light differs significantly from that predicted by the flux-limited model in LILAC, and cross-beam energy transfer has been identified as accounting for additional scattered light. This work is described in Sec. X B (see Fig. 10-18).

The dominance of inverse bremsstrahlung as the primary absorption mechanism is encouraging for ignition-scale designs since the absorption increases with scale length. For example, the predicted absorption for the design of Sec. III A is 95%. However, it is important to recognize that laser–plasma interaction processes in the associated longer-scale-length plasmas can be competing mechanisms, with the potential to reduce the absorption. It is therefore critical that absorption experiments continue to be made as larger plasmas are encountered.

High-convergence, direct-drive implosions require extremely uniform irradiation to minimize the seeds for hydrodynamic instabilities. It is generally considered that the overall level of nonuniformity should be ≲1% rms (root mean square), but the details of the modal structure are crucial to determine the effect of nonuniformity on target performance. Shorter-wavelength nonuniformities that are imprinted on the target can grow rapidly during the acceleration phase of the implosion by the Rayleigh–Taylor instability and cause the target shell to break up. Longer wavelengths will grow more slowly, but they can feed through to the inner surface of the shell and continue to grow during the deceleration phase, disrupting formation of the hot spot. In addition, long-wavelength nonuniformities limit the attainable convergence. The level and spectrum of irradiation nonuniformity depend on a number of factors including the number of beams, beam placement around the target chamber, the lens f number, the beam intensity profile, the beam smoothing applied, and coronal plasma conditions that affect laser absorption.¹⁵⁹ Beam uniformity is also beneficial in reducing the amount of undesirable laser–plasma interactions.

Some processes involved in the calculation of irradiation uniformity (and the resultant uniformity of target drive) are illustrated in Fig. 6-1, taken from Ref. 159, which shows two of many overlapping beams at tangential focus, each beam irradiating half the target surface. Not shown in the figure, but included in the calculations, is the refraction of laser rays as they pass through the hot plasma atmosphere surrounding the target. After laser energy is deposited (which generally occurs close to the critical density), some of the nonuniformities are smoothed by thermal conduction as heat is transported inward to the ablation surface, where the implosion is driven. Shorter-wavelength nonuniformities are more easily smoothed. Therefore, in examining nonuniformity in laser deposition, considerable emphasis is placed on calculating the spatial wavelength of the nonuniformities in addition to their magnitude.

To estimate the magnitude and wavelength of nonuniformities from multiple overlapping laser beams, a simple model can be used in which all beams have identical, circular intensity profiles with all beam axes pointed toward the center of the target. (More-detailed calculations investigate deviations from these ideal conditions.) The nonuniformity in energy deposition can now be decomposed into spherical harmonics, σ_ℓ, as discussed by Skupsky and Lee,¹⁵⁹ with the geometrical contribution of the laser configuration clearly separated out

where the total rms standard deviation over all modes σ_rms is defined as

The term in square brackets in Eq. (6-1) is a geometrical factor determined by the number and orientation of the beams (Ω_k) and the beam energies (W_k). The sum is over all beams, W_T= ΣW_k, and P_ℓ is a Legendre polynomial. The remaining factor on the right-hand side $|Eℓ/E0|$ contains all the information about energy deposition by a single beam: focus position, f number, beam profile, and target conditions. The beam-smoothing techniques described in Secs. VI B–VI D are designed to reduce the E_ℓ.

Figure 6-2 (from Ref. 160) shows an example of how direct-drive uniformity depends on the number of overlapping laser beams uniformly distributed around a sphere. Each beam is treated as a converging cone whose tip is at the focal point of the lens, with the axis of each beam passing through the center of the target. The total rms nonuniformity σ_rms is plotted as a function of focus ratio for several beam configurations, all subtending a total solid angle of 2% of 4π. The focus ratio is defined as $Rbeam/Rtarget,$ where R_beam is the radius of the beam at the target location and R_target is the initial target radius. To a good approximation, a focus ratio of 1 corresponds to tangential focus (when the edge rays of the beam are aimed at the edge of the target). The beam profile was assumed to vary quadratically with radius r as 1−(r/R_beam)². Only energy deposited between critical density and 1/3 critical was kept in the model and projected onto the surface of the sphere. Individual beam nonuniformities were not included. At the onset of irradiation, the focus ratio is generally chosen to be 1. During the time of irradiation, the target could implode to about 60% of its initial radius, resulting in a focus ratio of ∼1.7 at the peak of the laser pulse for a constant laser spot size. Low levels of nonuniformity (σ_rms ≲ 1%) are required over the entire range of focal conditions. Some laser systems employ “zooming,” wherein the laser spot size is decreased as the target implodes; in this case, the focus ratio would remain approximately constant during the implosion. For a given focus ratio, the exact level of uniformity depends on the details of single-beam laser absorption in the target corona, but the qualitative dependence on the number of beams can be expected to be as shown in Fig. 6-2.

The possibility of obtaining a high level of uniformity with a small number of beams was considered by Schmitt.¹⁶¹ He showed that perfect uniform illumination is possible with many aiming geometries and as few as six beams, provided that the absorption profile of each beam is proportional to cos²θ, where θ is the angle between an incident ray and the target normal at the aim point of the ray on the target surface. This model treats the absorption profile as a function of the angular coordinates on the sphere, neglecting the dependence on radial distance from the target surface. In reality, energy is deposited continuously along ray paths, with energy from rays near the edge of the beam (which refract through the plasma toward lower densities) being deposited at larger radii (see Fig. 8-20 below for typical ray trajectories). Schmitt and Gardner¹⁶² included refraction and a model of the ablation pressure that took into account the radius at which energy was deposited. They investigated the 32-beam configuration based on the faces of the truncated icosahedron that was proposed by Howard¹⁶³ and used in Fig. 6-2. Howard reported beam overlap calculations for 4–20 beams.

Figure 6-2 shows that good uniformity can be obtained with 32 beams, but this will occur over a limited range of laser–plasma conditions and will be very sensitive to variations among the beams such as pointing errors, energy imbalance, and spot shape variations. Moreover, Fig. 6-2 does not take into account the evolving plasma conditions. To maintain high levels of uniformity over a broad range of laser and target conditions, a 60-beam “stretched soccer-ball” configuration (based on the vertices of a truncated icosahedron)¹⁶⁴ w