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Laser Wake Field Collider

Recently NAano-Plasmonic, Laser Inertial Fusion Experiments (NAPLIFE) were proposed, as an improved way to achieve laser driven fusion. The improvement is the combination of two basic research discoveries: (i) The possibility of detonations on space-time hyper-surfaces with time-like normal (i.e. simultaneous detonation in a whole volume) and (ii) to increase this volume to the whole target, by regulating the laser light absorption using nano-shells or nano-rods as antennas. These principles can be realized in an in-line, one dimensional configuration, in the simplest way with two opposing laser beams as in particle colliders. Such, opposing laser beam experiments were also performed recently. Here we study the consequences of the Laser Wake Field Acceleration (LWFA) if we experience it in a colliding laser beam set up. These studies can be applied to laser driven fusion, but also to other rapid phase transition, combustion, or ignition studies in other materials.


I. INTRODUCTION
In recent years the Laser Wake Field Acceleration (LWFA) became a well known concept with useful applications. An intensive laser pulse impinging on a target creates a high density plasma of 4 x 10 19 / cm 3 , and a wake field wave follows the pulse. This, non-linear wave in dense plasma is formed of the EM-field, electrons and ions. A typical laser pulse of 20 mJ intensity, 7 fs length and λ wave length can create a Laser Wake Field (LWF) dense plasma wave of about 10 λ wavelength.
This wave is different from radio transmission waves in air or vacuum, where the material is dilute, not or weakly ionized. Still there are interesting phenomena if radio transmission waves create an interference. In the 1950s this radio wave interference was well known due to radio jamming (e.g. jamming the Radio Free Europe shortwave AM 49m band broadcast in Eastern Europe). Strong, unmodulated jamming broadcast on the same carrier frequency could lead to noiseless quiet sound, or it was white-noise modulated resulting in strong noise. Interference of two original frequencies that are quite close can lead to a "beat frequency transmission with f beat = (f 1 − f 2 )/2. This is often too low to be perceived as an audible tone or pitch, instead, it is perceived as a periodic variation in the amplitude of the broadcast. We aim to study similar kind of variety of possibilities in colliding LWF waves.
Laser Wake Field Collider (LWFC) waves can be realized the simplest way by two opposing laser light on a target. This was recently suggested in ref. [1] for laser driven fusion. Here two known effects were combined. First, the possibility of detonations on space-time hyper-surfaces with time-like normal, so called time-like detonations [3,4] which were found theoretically and experimentally in high energy heavy ion collision in the couple of last decades. This simultaneous volume ignition eliminates the possibility of Rayleigh-Taylor instabilities, which is a serious obstacle in achieving laboratory scale nuclear fusion. The second effect we use is to achieve simultaneous detonation in the whole volume of the target, by regulating the laser light absorption with nano-shells or nano-rods [5,6].
Using these two ideas a one-dimensional fusion configuration was suggested in ref. [1], with two opposing energetic laser beams.
Previous works also used colliding laser beams and LWFA for electron acceleration. In ref. [7] with a gas-jet target, the primary stronger beam created the LWF bub- The electric field, Ey (top) and magnetic field, Bz (bottom) in a Laser Wake Field (LWF) wave formed by irradiation from the ±x-direction. The rest number density of the H target is nH = 2.13 · 10 25 /m 3 = 2.13 · 10 19 /cm 3 . The laser beam wavelength is λ = 1µm. The LWF wavelength is about 20 λ.
A different method for electron accelerator was presented in ref. [9] where two identical laser beams irradiated an array of nanoantennas, which were nano-rods aligned with the polarization of the laser light. Electron bunches were hitting the layer of these nanoantennas for the side aligned with the directions of the nano-rods. These bunches were then accelerated each time passing a nano-rod with the right frequency and bunch period length. Although this proposal utilized nanoantennas, but for the local accelerating electric field to accelerating electrons. The target was a static nanoantenna array, changing its phase due to the incoming laser pulses.
Another experiment using colliding laser beams [10] aimed for achieving high target density for nuclear fusion, but did not utilize LWF waves and nano-plasmonics and did not attempt to have simultaneous transition in the target.
Thus, our present aim differs from these previous works.

II. NON-THERMAL IGNITION RATE
In Laser Wake Field Collider, the target has two sides, which are initially accelerated towards each other.
Carbon C6+ ions in the relativistic-induced transparency acceleration (RITA) regime, have reached near to 1 GeV energy on 300 nm target thickness [11].
In ref. [12] it was shown that intense laser pulse irradiating a combination target can accelerate Carbon ions to the TeV level by the laser plasma wakefield.
If we consider LWFC with a double layer target 1 precompressed to ion density, n pc , and pre-accelerated to several GeV/nucleon energy, (i.e. to a Laser Wake (LW) velocity near to the speed of light, v LW ≈ c), the two LWF waves can inter-penetrate and lead to an ignition reaction rate of where σ is the ion-ion cross section, and due to the Lorentz contraction of the two ion bunches are compressed to γ n pc . This may well exceed the thermal (th) rate of n v th σ . If the ions are accelerated to 5 GeV/nucleon, then γ ≈ 6, and if the pre-compression reaches a factor 8 (considerably less than at NIF, where the 3-Dim compression reaches 800 g/cm 3 ), then our burning rate is 2 γ 2 8 2 n 2 800 n (c/v th ) ≈ 270 times bigger than at NIF. Here we assumed that the average thermal collision speed is v th ≈ c/2. So, the non-thermal nonequilibrium Laser Wake Field Collider mechanism may well exceed the thermal ignition rate by the adiabatic compression and heating at NIF. Especially if this ignition takes place for a central hot-spot only and then the flame should propagate over the rest of the target.
The other advantage of the LWFC configuration that we can have exclusively the most optimal DT reactions, without DD and TT ones, like in a mixed DT target.

III. LASER AND TARGET PARAMETERS
Irradiating a dense target with such beams creates LWF waves, and we study what are the consequences. We used the EPOCH multi-component PIC code to see first what kind of LWF waves develop, if we irradiate a target with a laser beam of wavelength λ = 1µm, a Gaussian distribution in the transverse, [y, z] plane of half amplitude time δ t = 26fs, and full pulse length ∆ t = 52fs.
The laser focus average diameter is 2R = 40µm. The laser pulse energy in the present test is 19.6 J, and the maximum intensity, in the center of the transverse plane at the top intensity time is 3.0 · 10 19 W/cm 2 . We studied the EM filed and the target in a 3dimensional box of 50 x 70 x 70 µm, divided into cubic Cells of 100 nm (0.1 µm). The initial target consisted of H atoms. The H target consisted two slabs of 19.5 µn thickness with a 1 µn gap between them, and preaccelerated towards each other with a momentum og 100 MeV/c. Due to the irradiation the target became mostly ionized, and protons p and electrons, e are formed. These components were represented by marker particles. The rest number density of H atoms in marker particles in this work is n H = 2.13 · 10 27 /m 3 = 2.13 · 10 21 /cm 3 , or n H = 2.13 · 10 25 /m 3 = 2.13 · 10 19 /cm 3 . This is near to the density of liquid hydrogen, which is 2.124 · 10 22 /cm 3 . The initial size of these marker particles is 0.025 3 · µm 3 = 1.5625 · 10 −5 µm 3 . For comparison we also made a test with a two orders of magnitude more dilute target. The different types of marker particles (mp) contained different number of components: 1e mp ∝ 10 24 e, 1p mp ∝ 10 24 p, 1H mp ∝ 10 24 H, Initially the H component marker particles were uniformly distributed in the calculation cells, so that each PIC Cell contained 4H mp . In Fig. 2 we see the effect of two opposing laser beams on the two target H atom slabs. The two beams hit the denser targets from opposite sides, and as the irradiation is absorbed the H atoms are ionized, protons and electrons become free. By 150 fs, the H atom targets are fully ionized, except at the outside edges where the pinched irradiation beams did not hit the target.
Electrons were created during the irradiation. H ions and electrons were moving during the calculation and crossed over to other Cells.
In Fig. 3 we can see that at around 125 -150 fs at this lower target density, the two LWF waves constructively interact and the EM field strength is maximal. This moment of time would be adequate for a short, intensive ignition pulse.
The EM field strength for higher initial target densities, n H = 10 21 /cm 3 , decreases strongly and becomes random. This is the consequence of the strong absorption by the denser target. We can see the signs of increasing randomness in Fig. 4. The amplitude of both E y and B z is reduced and random fluctuations increase. This can be seen already in Fig. 4, where at 150 fs, the penetration of the B z field into the target is delayed and reduced, and the constructive interference the two opposing B z fields is delayed.
This can be attributed to the kinetic approach, where the interactions and pressure in the target is neglected and thus the dominant longitudinal momentum is transported to the kinetic motion of the target particles.
These then contribute to the pinch effect reducing the EM strength and target beam directed momentum.
The model includes a kinetic collision setup, which reproduces the relaxation time approximation, so that the momentum distribution converges towards the Maxwell-Boltzmann ideal thermal distribution. This can then be characterized by a temperature. At the same time this effect is demonstrated for collision within the same type of particles.
Thus it is more realistic for our modeling to consider the target and projectile Atoms and Ions as separate particle species and different type of marker articles (simulation particles: H p , H t , p p , and p t , while the electrons may remain in the in a single group, as these are more spread out in the space due to their smaller mass.
The LWF waves show increasing penetration into the target with increasing laser beam energy. This is also reconfirmed by recent experiments [13], showing that with increasing laser beam energy the reflection of the beam decreases, while the absorption increases to 100% in case of Au target.
One should check actually the energy density distribution and its time dependence, in this dynamical situation. Up to now in the target studies [6] mainly static final state configurations were discussed.
As the pressure and the Equation of State does not appear in the EPOCH code, it is clearly solving the kinetic equations of motion for the electrons and ions. Relativistic thermal distribution functions are included in the code.  Fig. 3 with target density nH = 2.13 · 10 21 /cm 3 . At this higher density the energy of EM radiation is absorbed by the ionization of the target.
In dense nuclear plasma, the EoS is also vital, as at lower densities we have nuclear attraction while at higher densities strong nuclear repulsion. In ideal gas kinetic theory these essential effects are not taken into account. This could play a significant role in correct estimate of the burning reaction rate of the target.
An experimental reconfirmation of the effect of energetic laser beams was done on a Gold target [13], recently, and strong light absorptions as well as the conversion effect to X-rays was observed. This experiment actually confirms earlier, original ideas in [1,2,4].
In [13], the conversion to X-rays radiation is parametrized by a "velocity", attributing the frequency shift to a Doppler effect. Nevertheless, this velocity is not connected to any of the several possible and above mentioned processes, and the authors do not elaborate what this velocity parameter would characterize.

V. OUTLOOK FOR NANOANTENNAS
The penetration of the LWF waves into the target was mentioned earlier, and in colliding beam configuration this leads to substantial ion density increase. The role of nanoantennas in this case is that the increased photon absorption leads to increased momentum deposition in the target and higher density [1]. Finally this results in faster burning rate.
The nanoantennas have another effect too. Similarly to the golden (or depleted Uranium) hohlraum, which converts the incoming laser light, to X-rays. In case of internal nanoantennas in the target DT fuel we have two main effect for conversion of the visible laser light to higher X-ray frequencies: Bremsstrahlung in electron collisions, and transition from high energy level electron states to lower ones. This second effect is especially strong for resonant nanoantennas, which lead to periodic extreme high electron densities at the edge of the antenna. As the electrons are Fermions, at high density they are forced to occupy high energy levels (at the cost of the incoming laser energy) and then as the density periodically decreases they emit the corresponding X-rays. This process is specific to nanoantennas. Due to momentum conservation the incoming laser light momentum and the emitted X-ray momentum are relatively aligned.
A recent experiment shows [13], that even without nano-antennas, laser light with increasing intensity on gold target leads to increasing absorption (and vanishing reflection). Furthermore, at the same time the increasing intensity laser irradiation is accompanied by increasing xray conversion. Embedded resonant nanoantennas are expected to amplify these effects. Resonant nanoantennas have an electron density fluctuation parallel to the change of the EM field of the laser irradiation. At 1.24 · 10 18 W/cm 2 laser beam intensity this is estimated to be of the order of 700 C/m 2 [6]. If we assume that the electron plasmonic wave has a thickness of about 1-10 nm, then this surface layer has an electron charge density of 7 − 70 · 10 10 C/m 3 or 4 − 40 · 10 29 C/m 3 , which is larger than the electron charge density arising from the laser wake, as shown in Fig. 5. This indicates that the electron density increase due to nanoantennas will significantly contribute to the conversion to X-rays.
The Bremsstrahlung effect is always present in electron collisions with, atoms, ions and other electrons. Also electron collisions on nano particles can be taken into account as an additional effect of nanoantannas. More importantly at the point when all atoms are ionized the nanoantannas do break apart also with dynamical domains of high electron density. These domains lead to additional collisions and Bremsstrahlung. Thus the internal resonant nanoantennas have an increased conversion of visible light to X-rays.
These several effects connected to nanoantenans are quite complex, thus a a precise, quantitative theoretical evaluation is not a real possibility.
Thus, it is important to study experimentally, how the LWF waves and the nanoantenna distribution can achieve the best time-like ignition in most of the whole target! Interactions of electrons on the nano-plasmonic surfaces, which can interact with the surrounding H ions may lead eventually to e(p, n)ν e reactions in case of high electron density. The low energy neutrons then can form Deuterium ions, H + (n, γ)D + . See: https://wwwnds.iaea.org/ngatlas2/