Elliptical Magnetic Mirror generated via Resistivity Gradients for Fast Ignition ICF

The elliptical magnetic mirror scheme for guiding fast electrons for Fast Ignition proposed by Schmitz (H.Schmitz et al., {\it Plasma Phys.Control.Fusion},{\bf 54} 085016 (2012)) is studied for conditions on the multi-kJ scale which are much closer to full-scale Fast Ignition. When scaled up, the elliptical mirror scheme is still highly beneficial to Fast Ignition. An increase in the coupling effiency by a factor of 3--4 is found over a wide range of fast electron divergence half-angles.


I. INTRODUCTION
In this paper we make a first assessment of a fast electron guiding concept proposed by Schmitz et al. 1 which exploits resistivity gradients 2 to self-generate an elliptical magnetic mirror which in turn helps collimate a beam of fast electrons with a large angular divergence.
The original proposal by Schmitz only considered conditions, spatial, and temporal scales that were considerably different to full scale Fast Ignition inertial fusion.Here we attempt to bridge this disparity and thus make an improved assessment of the concepts utility in Fast Ignition.
Fast Ignition (FI) 3,4 is a variant of inertial confinement fusion (ICF) in which the compression and ignition of the target take place in two separate stages.The ignition of the target is done by using an extremely high power driver to produce a beam of energetic particles (this could be multi-MeV electrons or ions) to generate a hot spot in the compressed fuel.Compared to central hot spot ignition, it has the advantages of higher gain for lower total laser energy (a total energy of a few hundred kJ may be possible 5 ).FI concepts based on fast electrons usually make use of a reentrant cone to shield the path of the ultra-intense laser driver from coronal plasma.There will be a considerable stand-off distance between the apex of the cone and the centre of the compressed fuel.A crucial element of the scheme is then the efficient, i.e. well collimated, transport of the fast electrons from the cone apex to the hot spot.The radius of the laser spot and the hot spot are likely to be comparable, but both are smaller than the stand-off distance by a factor of a few, so the beam must be fairly well collimated to ensure efficient coupling.If the angular spread in the fast electrons is small, then the resistive generation of a collimating magnetic field may be sufficient to pinch the beam and provide such collimation.Considerable effort has been put into modelling the fast electron transport aspect of FI, some of which can be found in [6][7][8][9][10][11][12] and references therein.
Extensive experimental and simulation studies seem to indicate that the angular spread of the fast electrons under conditions relevant to both FI and laboratory-scale condtions may be quite large 6,[13][14][15][16][17] .This means that any resistive magnetic field generation in the DT fuel will be insufficient 18 to significantly pinch the beam, and low coupling into the hot spot will result.The problem of 'fast electron divergence' has motivated a considerable research effort into modifications of the reentrant cone FI concept, and a number of approaches to addressing this problem have been put forward 2,[19][20][21][22][23] .
Here we show that the Elliptical Magnetic Mirror concept proposed by Schmitz can be highly beneficial in terms of mitigating the loss of coupling efficiency caused by large fast electron divergence even when one extends one's considerations to conditions much closer to full-scale FI than the original description of the concept.

II. ELLIPTICAL MIRROR CONCEPT
The elliptical mirror concept 1 is based on the idea of magnetic collimation, first proposed by Robinson and Sherlock 2 , and later developed in further theoretical and experimental 24,25 work.The electrons are collimated by azimuthal magnetic fields which are generated by resistivity gradients inside the target.One can assume both charge and current neutrality, i.e. j f + j b = 0, where j f is the fast electron current and j b is the background electron current.Calculating the background current through Ohm's law, ηj b = E, where η is the conductivity, Faraday's law can be written as 26 The first term on the right-hand side generates a magnetic field that directs electrons towards regions of higher current density, and thus acts to collimate the fast electron beam.The second term, which forms the basis of the elliptical mirror concept, generates a magnetic field at resistivity gradients which acts to keep the fast electrons within regions of higher resistivity.The resistivity gradient is created by a transition between two materials with different Z.The high energy electrons are generated within the high-Z material and the magnetic field at the material interface will deflect the electrons and keep them inside the high-Z domain.
For sufficiently steep resistivity gradients one can assume that the electrons are reflected specularly off the magnetic layer.In this case the electrons feel the magnetic field only for a short time and move only a short distance along the tangential direction.If the magnetic field in the layer does not vary along the tangential direction over this short distance then the electron trajectory is symmetric with respect to the normal direction and the electrons are specularly reflected.This can be used to construct geometries for collimating electrons similar to optical mirrors.In this approximation, an elliptical geometry with one focal point in the injection region and the other focal point in the hot spot should be highly effective at focusing the electron beam into a hot spot of roughly the same size as the laser spot.

III. SIMULATIONS A. Set Up
Simulations were performed using the 3D particle hybrid code zephyros 24,25,27 .This is based on the hybrid method developed by Davies in a series of publications 26 The main parameters varied in these simulations (designated A-G) were θ div , and r spot .We also considered the case of a combination of higher intensity and sub-ponderomotive fast electron temperature scaling.
A number of control runs (designated CMP1-CMP4) were also carried out to examine the case where the elliptical mirror is not present.In the control simulation only the cone tip and the compressed fuel were present in a 200×200×200 box.All other simulation parameters were same as the base-line.A table summarizing the parameters used in the different runs is shown below (table I).
The geometry of the Al semi-ellipsoid is chosen so that the centre of the injection region sits at one of the foci of the (full) ellipsoid, and so that the target 'hot-spot' sits at the other focus.As can be seen from figure 1, the minor axis of the ellipsoid, b, is 30 µm.The major axis, a, was therefore chosen to be 104.4µm (from f = √ a 2 − b 2 ), so that the foci of the ellipsoid sit at the desired positions.The quality of guiding obtained by this configuration rests on both establishing a strongly reflecting magnetic field at the ellipsoid surface, and on the aberration that arises from having an extended source being negligible.

B. Baseline Simulation
In the baseline simulation we can see a number of the general features by looking at the magnetic field (B z component) and fast electron density in the x-y mid-plane some way into the simulation.These are shown at 4 ps in figure 2.
Simulation θ div r spot (µm)    The baseline simulation is also shown in figure 4. Without the elliptical mirror, the coupling efficiency is 5.2%.So the elliptical mirror increases the coupling efficiency by a factor of 3.7.So, the baseline scenario, in which the angular spread of the fast electrons is quite considerable anyway, indicates that the elliptical mirror is highly effective.Note that the coupling efficiency obtained in the baseline simulation is greater than in all the comparator simulations (CMP1-CMP4).

Fast Electron Divergence Angle
In simulations A-E we have extended the analysis of the baseline simulation, to look at a wider range of fast electron divergence half-angles.The coupling effiencies that were extracted, including simulation A, are shown in fig. 5 alongside the results from CMP1-CMP4.It can be seen that although the coupling efficiency falls with increasing divergence half-angle, which is entirely expected even with the mirror, the coupling efficiency is always at least 3 times higher with the mirror than without.We therefore find that the elliptical mirror can be highly beneficial across a range of fast electron divergence angles.

Fast Electron Source Radius
In simulations F and G we varied the radius of the fast electron source region.In simulation F, the source radius was 14µm (compared to 15µm in A-E).The total injected fast electron energy was 20.07 kJ and the energy coupled into the hot spot was 4.2 kJ, i.e. a 20.9% coupling efficiency.In simulation G, the source radius was 13µm (compared to 15µm in A and B).In G the total fast electron energy was 17.33 kJ, and the energy coupled into the hot spot was 3.07 kJ, i.e. a 17.7% coupling efficiency.There is, therefore, some slight dependence on the laser spot radius.One might expect that some optimization can be performed by tuning the hot-spot radius, because as the hot spot radius is reduced the fast electron source becomes more point-like, and the elliptical mirror configuration is able to focus this into a smaller volume.These runs show that the improvement that can be gained is only likely to be a little.If the hot spot radius is reduced too much then there won't be enough energy to achieve ignition, so the extent of this optimization is very limited.
The poor performance observed in simulation G appears to be due to magnetic fields that develop in the interior of the Al ellipsoid and which produce a strongly annular transport pattern.This is illustrated by figure 6 below which shows the fast electron density at 4 ps in run G.The formation of such 'interior' magnetic fields is consistently seen throughout the simulations (see figure 2, where it is clearly visible), and is an obvious consequence of the fast electrons not uniformly filling the ellipsoid as they propagate along its length.
In simulation G these fields have developed much more strongly leading to highly annular transport.The annular transport leads to a ring-like heating pattern in the DT, hence the coupling efficiency is mainly reduced energy being deposited further out radially.Clearly this sort of behaviour can occur in this guiding geometry where the ellipsoid is relatively long (a ≫ b), and future work may have to examine ways to mitigate this problem (although it is currently only observed in a minority of cases).

D. Effect of Alternative Fast Electron Scaling
In simulations A-G, only a few kJ are coupled into the hot spot.Partly this is a consequence of the fast electron scaling with laser intensity (pondermotive).In previous studies of Fast Ignition energy coupling, it has been noted that a ponderomotive energy scaling is not conducive to Fast Ignition.This is because the fast electron range is approximately given by ρR = 0.6T (MeV)gcm −228 , so the ideal mean fast electron energy lies in the range of 1-2 MeV.The ponderomotive scaling therefore tends to force one to have an unfavourably high (>2 MeV) mean fast electron energy in order to inject sufficient energy in under 20 ps.Some studies of fast electron generation, on the other hand, have observed what might be termed 'sub-ponderomotive' fast electron scalings 29,30 which, if realizable at full scale, are more conducive to Fast Ignition for the aforementioned reasons.The matter of fast electron scalings is still a subject of on-going research, and the realization of such a scaling under full Fast Ignition conditions may not be possible.Nonetheless, we carried out further simulations to see if the coupling efficiencies are maintained under these different conditions.
The further simulations (H-K)made the following modifications to the baseline simulation: (i) laser intensity was increased to 6×10 20 Wcm −2 , (ii) T f =0.6T p , (iii) conversion efficiency was decreased to 40%.Otherwise the simulation parameters are unchanged.The particular choice of T f =0.6T p is taken from the results obtained by Sherlock 31 .The total injected fast electron energy is 27.65 kJ, so the energetic scale of these simulations is comparable to the others in the series.The actual value of T f is 1.95 MeV The analysis of the energy coupling into the cubic 'hot spot' is identical to that done previously.One extra comparator run (CMP5) was performed.The simulation results in terms of the coupling efficiency are shown in figure 7 below.In the equivalent ponderomotive comparator, CMP4, the coupling efficiency was 9.5%.
As CMP5 has a 13% coupling efficiency, the effect of lowering T f has been to improve the coupling efficiency by about 37%.The coupling efficiencies with the elliptical mirror are higher than in simulations A-E (see figure 5), but this largely expected from the increase in coupling due to lowering T f .The variation of coupling efficiency with the fast electron divergence half-angle is flatter overall From figure 7 it can be seen that the elliptical mirror configuration still produces very good coupling efficiencies into a realistically sized hot spot over a wide range of divergence half-angles, and that these coupling efficiencies are better than what is achieved without any guiding scheme by a factor of a few.
. A 250×200×200 grid was used with a 1µm cell size in each direction.The target set up is shown (in terms of atomic number and mass density) in figure 1, and these profiles are axisymmetric about the y = z =100µm line.The target consists of Al re-entrant cone the top 100 µm of which contains a guiding structure consisting of a truncated semi-ellipsoid core of Al surrounded by a CH 2 substrate.Outside the core is a mass of compressed DT, the centre of which is centred at r DT =(200,100,100) µm.The radial density of the fuel profile is ρ = 10 + 400 exp [−(r/R) 4 ]gcm −3 , with R =50µm.The left hand edge of the simulation domain represents the edge of the insert in the cone tip.To the left of this there will only be vacuum, and the laser irradiates this surface.

FIG. 2 .
FIG. 2. (a)log 10 fast electron density (in m −3 ) at 4 ps in baseline simulation (A) in x-y mid-plane of simulation box.(b) B z (T) component of magnetic field at 4 ps in baseline simulation (A) in x-y mid-plane of simulation box.

Fig. 2 (
Fig.2(a)shows that the fast electrons propagate from the cone tip into the dense fuel with relatively little transverse spread, as one would have hoped the mirror scheme to work.By comparing figures 1 and 2, it can also be seen that the fast electrons are very well confined to the semi-ellipsoidal Al guide element.The magnetic field plot, fig.2(b), shows that a strong elliptical mirror has been generated at the interface between the Al guide structure and the CH substrate in the cone tip insert.Clearly it this azimuthal magnetic field structure that is providing the excellent confinement observed in the plot of the fast electron density.Also note the formation of some magnetic field within the ellipsoid close to the axis, which is due to inhomogeneous propagation of the fast electrons.These magnetic fields may be an inhibiting factor in achieving the best performance from this configuration, and we return to

FIG. 6 .
FIG. 6. (log 10 fast electron density (in m −3 ) at 4 ps in simulation G in x-y mid-plane of simulation box.