Inertial fusion energy (IFE) is based on high compression^[1–3]. The reasoning is that it is energetically cheaper to compress rather than to heat and the nuclear reactions are proportional to the density squared. IFE of deuterium–tritium (DT) requires high compression () and, in particular, the aneutronic fusion^[4–6] of proton–boron11 needs extremely high compression (). The high compression is achieved by shock waves and the accumulation of matter during stagnation of the implosion of the target shell.

Shock waves in laser plasma interactions^[7] have played an important role in the study of IFE. In 1974 the first direct observation of a laser-driven shock wave was reported^[8]. A planar solid hydrogen target was irradiated with a 10 J, 5 ns, Nd laser and a pressure of approximately 2 Mbar was measured. Twenty years after this experiment, the Nova laser from Livermore created a pressure of Mbar^[9]. This was achieved in a collision between two gold foils, where the flyer (Au foil) was accelerated by a high intensity X-ray flux created by the laser–plasma interaction.

In order to achieve nuclear fusion ignition, a mega-joule (MJ) laser with a few nanoseconds pulse duration has been constructed in the USA^[10]. The central spark ignition of DT is expected in the near future. In this scheme the target and the driver pulse shape are designed in such a way that only a spark at the centre of the compressed fuel is heated and ignited^{[11, 12]}. The rest of the fuel is heated by particles produced in the DT reactions.

In order to ignite a DT target with significantly less than a few MJ of energy, it was suggested^{[13, 14]} to separate the drivers that compress the target from those that heat the target. This idea is called fast ignition (FI), and triggers not in a central spark, but in a secondary interaction of an igniting driver of a very short duration, such as a multi-Petawatt (PW) laser beam. The PW laser is supposed to form a channel for a few picoseconds in the plasma atmosphere and to ignite a part of the fuel at the stagnation point of the implosion. For this purpose it is estimated that ignition requires of the order of a few tens of kilo-joule of laser energy for a duration of approximately 10 ps with an irradiance of the order of . The FI problem is that the laser pulse does not penetrate directly into the compressed target, which has an electron density of the order of . Therefore many schemes of FI have been suggested: (1) the laser energy is converted into electrons that ignite the target^[15]. (2) The laser energy is converted into protons that ignite the target^[16]. (3) Since the heating in the previous proposals is not confined and furthermore it is necessary to avoid preheating, a gold cone^[17] (Au density/solid DT density ) is stuck in the spherical pellet. (4) FI is induced by plasma jets^[18] that are induced by the same laser system that compresses the pellet. (5) The FI is achieved by a plasma flow created from a thin exploding pusher foil^{[19, 20]}. (6) Plasma blocks for FI have also been suggested^{[21, 22]}. (7) Murakami et al.^[23] revived the old idea of impact fusion with the help of the cone. (8) The use of clusters^[24] was also suggested to ignite the compressed pellet. (9) Furthermore, the use of an extra laser-induced shock wave created by the same lasers that compressed the target in order to ignite the target was suggested^[25]. (10) Alternatively the FI shock wave from a laser-accelerated impact foil^{[26, 27]} was proposed. The shock wave ignition schemes are actually based on heating by collision of two shock waves using tailored laser pulses that had already been suggested^[28] even before the idea of FI was explicitly published^{[13, 14]}.

It is well known that the interaction of a high power laser (HPL) with a planar target creates a one dimensional (1D) shock wave^{[29, 30]}. The theoretical basis for laser-induced shock waves analysed and measured experimentally so far is based on plasma ablation. For laser intensities in the range and nanosecond pulse durations a hot plasma is created. This plasma exerts a high pressure on the surrounding material, leading to the formation of an intense shock wave moving into the interior of the target. The momentum of the out-flowing plasma balances the momentum imparted to the compressed medium behind the shock front, similar to a rocket effect.

For the ablation pressure is dominant. For the radiation pressure is the dominant pressure at the solid–vacuum interface and the ablation pressure is negligible. In this last case the ponderomotive force drives the shock wave. For laser irradiances one gets a relativistic laser-induced shock wave^[31]. The theoretical foundation of relativistic shock waves is based on relativistic hydrodynamics^[32] and is first analysed by Taub^[33]. Relativistic shock waves may be of importance in intense stellar explosions or in collisions of extremely high energy nuclear particles. Furthermore, relativistic shock waves may be a new route for FI nuclear fusion.

In Section 2 the formalism of relativistic shock waves is given for further consideration. In Section 3 the laser-induced shock wave equations are explicitly written and solved numerically without approximation for the first time. In a recent publication^[31] the solution is given only for very strong relativistic shocks, whereas in this paper the transition between relativistic and nonrelativistic laser-induced shock waves is obtained. It turns out that this transition domain is important and relevant for the FI scheme as described in Section 4. The paper is concluded with a short summary and discussion.

The relativistic 1D (or non-relativistic^[34]) shock wave is described by five variables: the particle density (or the density , where is the particle mass), the pressure , the energy density , the shock wave velocity , and the particle flow velocity , assuming that we know the initial condition of the target: (or ), , , and the particle flow velocity , before the shock arrival. The four equations relating the shock wave variables are the three Hugoniot relations describing the conservation laws of energy, momentum, and particles, and the equation of state^{[35, 36]} connecting the thermodynamic variables of the state under consideration. In order to solve the problem an extra equation is required, which in our case we derive from a laser–plasma interaction model.

The relativistic hydrodynamic starting point is the energy momentum 4-tensor given by

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

From equations (6) the velocities and are obtained

 (7)

 (8)

Assuming that in the laboratory the target is initially at rest, , the shock wave velocity and the particle flow velocity in the laboratory frame of reference are related to the flow velocities and in the shock wave rest frame of reference by

 (9)

 (10)

This paper analyses the shock wave created in a planar target by the ponderomotive force induced by very high laser irradiance. In this domain of laser intensities the force acts on the electrons that are accelerated and the ions that follow accordingly. This model describes our piston model^{[37, 38]} as summarized schematically in Figure 2: Figure 2(a) shows the capacitor model for laser irradiances , where the ponderomotive force dominates the interaction. In Figure 2(b) the system of the negative and positive layers is called a double layer (DL), and are the electron and ion densities respectively, is the electric field, is the distance between the positive and negative DL charges, and is the solid density skin depth of the foil. The DL is geometrically followed by a neutral plasma where the electric field decays within a skin depth and a shock wave is created. The shock wave description in the laboratory frame of reference is given in Figure 2(c). This DL acts as a piston driving a shock wave^{[39, 40]}. This model is supported in the literature by particle in cell (PIC) simulation^{[39, 41]} and independently by hydrodynamic two-fluid simulations^{[21, 22, 42]}. The relativistic shock wave parameters, such as compression, pressure, shock wave and particle flow velocities, and temperature are calculated here for any compression for the first time in the context of relativistic hydrodynamics. In a recent previous paper this was solved only for with .

For the ablation pressure is dominant and scales with the laser irradiance like , where is of the order of 2/3 in a 1D model^[7]. For the radiation pressure is the dominant pressure at the solid–vacuum interface and the ablation pressure is negligible. In this last case the ponderomotive force drives the shock wave. The equations forrelativistic hydrodynamics with the ideal gas EOS in the laboratory frame of reference are

 (11)

 (12)

 (13)

 (14)

 (15)

The numerical solutions of equations (11) and (12) are shown in Figures 4 and 5. Figure 4 gives the dimensionless shock wave pressure versus the dimensionless laser irradiance in the range . For a better understanding of this graph and for the practical proposal in the next section, the inserted table shows numerical values in the range . Figure 5 describes the dimensionless shock wave velocity and the particle velocity in the laboratory frame of reference versus the dimensionless laser irradiance in the range while the inserted table shows numerical values in the range . As a numerical example we take a target (liquid DT) with initial density irradiated by a laser with intensity , namely . In this case our relativistic equations yield a compression , a pressure bar, a shock wave velocity and a particle velocity , where is the speed of light.

The relativistic speed of sound for an ideal gas EOS is

 (16)

 (17)

We now analyse the temperature problem. The partial pressures of an ideal gas that contains electrons and ions with appropriate densities and and temperatures and are and , and can be described by

 (18)

 (19)

 (20)

 (21)

 (22)

 (23)

 (24)

Taking the example given above for liquid DT with and initial density irradiated by a laser with intensity , namely , we get and a temperature in the range . However, for we have electron–positron pair production^{[43, 44]} and new physics is required here for the temperature calculations. It is out of the scope of this paper to analyse this exotic case here.

In order to solve the energy problem of future generations scientists have considered using controlled nuclear fusion energy. One of the approaches is the well-known inertial confinement fusion driven by HPLs where the physics is based on compressing and igniting rather than confining the fuel^{[1, 2]}. In order to ignite the fuel with less energy it was suggested to separate the drivers that compress the target from whose that ignite the target^{[13, 14]}. First the fuel is compressed, then a second driver ignites a small part of the fuel while the alpha particles created in the DT interaction heat the rest of the target. This idea is called FI. The problem with FI is that the laser pulse does not penetrate directly into the compressed target; therefore many alternative schemes have been suggested^[45].

The laser solution of the energy problem requires very sophisticated high power laser science and engineering (HPLSE). In a recent paper^[46] the various HPLSE optimizations and design constraints for a laser fusion power plant are beautifully summarized and analysed. From the many possible proposals to solve the energy problem with HPLs we consider three criteria for choosing the best candidate (present or future): (i) Understanding the physics. In HPL–target interactions there are many scientific problems not yet fully understood, such as laser–plasma instabilities, hydrodynamic instabilities, equations of state, nonlinear transport issues, non-local thermodynamic equilibrium, even neglecting energy conservation! (ii) Engineering simplicity. The IFE project is extremely complicated technologically and therefore a major effort is required in choosing the laser system, target design, etc., from all possible proposals by physicists. Technological simplicity must be seriously taken into account. For example, IFE requires or more laser shots per year; therefore complicated target designs (such as inserting a golden cone inside a pellet) are not realistic. (iii) Last, but not least, IFE is supposed to be economically practical. This implies the required gain, defined as the nuclear energy output divided by the laser input per shot, be larger than 100 and that the cost of a target should not be more expensive than 0.1 US$.

Taking into account these three criteria it looks as though: (a) direct drive is simpler than indirect drive. (b) FI needs significantly less energy (approximately 0.3 MJ instead of 3 MJ). Therefore direct drive FI has the potential to be the best route to achieve nuclear fusion as an energy source. (c) From all presently known FI schemes the simplest FI seems to be by means of an ‘extra shock’ wave. We suggest a novel shock wave ignition scheme requiring less energy (in comparison with the present shock wave ignition scheme^[25]) and free of laser–plasma instabilities (no more than in the laser compression pulses). In this proposal the ignition shock wave is created by a high irradiance laser and the shock wave is induced by a ponderomotive force in the intermediate domain between the relativistic and non-relativistic hydrodynamics. For this case the relativistic shock wave formalism has to be considered as developed in our previous section. We call our scheme ultrafast, since the laser pulse duration for the ignition process is significantly smaller, by one to two orders of magnitude.

The shock wave ignition criteria for DT nuclear fuel are

 (25)

The compression of a typical pellet as discussed in the literature^{[12, 47]} requires between 100 and 300 kJ of energy, depending on the EOS, target design and the final required density. The FI in our case needs approximately 30 kJ of energy. Such a laser is under development and may be available in the near future.

Recently^[31] it was suggested that relativistic shock waves with a shock wave velocity of more than 50% light speed can be created in the laboratory with HPLs which are recently under development. In this paper we discuss two novel ideas. The first is the transition domain between the relativistic and non-relativistic laser-induced shock waves. The second is the use of this transition domain for shock-wave-induced ultrafast ignition of a pre-compressed target. The laser parameters for these purposes are calculated and the main advantages of this scheme are described. The many laser beams with few nanosecond pulses that compresses the target do not require as in the previously proposed shock wave ignition scheme^[25], thus disturbing laser plasma instabilities do not occur. Furthermore, in the present scheme less energy is required in the main laser pulses where a picosecond laser with very high power (30 PW) is required for the ultrafast ignition with the shock wave in the intermediate domain between the relativistic and non-relativistic hydrodynamics.

Presently existing petawatt lasers (see appendix B) might be used to start relativistic experimental research in the laboratory. Recent and future developments of HPLs in the multi-petawatt domain could be important for relativistic shock waves in the laboratory with pressures of atmospheres or energy densities of the order of . Such pressures or energy densities have been suggested so far only in astrophysical objects.

The ultrafast ignition scheme suggested in this paper appears advantageous in comparison with the many FI proposals, as given in our introduction section. It is based on the following merit criteria: (i) Understanding the physics, (ii) Engineering simplicity, and (iii) Economically practical. We think that shock wave FI is the best choice and the model suggested here between the relativistic and non-relativistic domain has significant advantages and should be taken seriously into account.

Finally we must mention a very recent FI proposal^[48] using a laser system similar to our laser parameters estimated in Section 4. In particular this scheme requires a temporally tailored pulse with an energy of 65 kJ of duration 1.48 ps with a maximum intensity of . This model is based on the use of a hole–boring^{[39, 49, 50]} phenomenon that enables the HPL beam to penetrate beyond the critical density. As early as 1971 it was analytically calculated^[49] that the condition for a laser to propagate in non-uniform plasma with an electron density and density gradient scale length beyond the electron critical density is given by

 (26)

In our model the laser-induced relativistic shock wave induces the ignition. The preliminary compressed fusion target for ICF is usually spherically symmetric and the density increases very rapidly towards the core of the target when our shock wave model is applied as described in Figure 7. As long as the thickness of the shock wave is much smaller than the density gradient , i.e., , we can assume that the target is uniform in the shock wave domain. Looking at our Table 1 one gets , which is much smaller than the density gradient derived in PIC simulations^{[48, 50]}.

The solution suggested in this paper, like all other solutions to the energy problem, is extremely difficult scientifically, and a lot of money and enormous optimism is required for a positive solution. HPLSE is complex, complicated but possible. If civilization is to survive we need large new sources of energy. To quote Mark Twain (1835–1910): ‘And what is a man without energy? Nothing – nothing at all’.