Abstract
1. Introduction
Laser-driven ion acceleration has created enormous interest over the last few years[
The observation of ions emitted in laser–plasma interactions can first be traced to experiments employing high-intensity laser pulses with durations of a few ns to some hundreds of ps[
Although a number of high-power, PW-class laser systems have been built around the world, the early record energy of 60 MeV[
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2. Target normal sheath acceleration
TNSA of ions has been extensively investigated over the last decade. In fact, until now it has proved to be the most effective method for ion acceleration when highly intense laser pulses are focused onto foils with thicknesses of several micrometers. TNSA relies on the efficient conversion of laser energy into hot, relativistic electrons. These electrons propagate through the target and set up fields at the target boundaries where they exit into vacuum. The electric field is created between the expelled electrons and the surface charge that they induce on the target. Since the electric field strength is of the same relative strength as the laser electric field amplitude (TV/m) which generates the hot electrons, most of the electrons return back into the target. Hence an electron cloud (sheath) is formed. Ions at the rear surface can be accelerated by the sheath fields to multi-MeV energies in only several tens of femtoseconds. Impurities such as hydrocarbon and water are present under most experimental conditions at the solid surface. Due to their higher charge-to-mass ratio, it is the protons from these contaminants that are most readily accelerated to high energies. However, by removing the contaminants by different means, the acceleration of heavier ions can also be optimized[
A large number of experimental results on laser-driven ion acceleration are now available[
2.1. Nonrelativistic TNSA
The original Schreiber model[
In the following we will concentrate on the center of the acceleration region that is responsible for the most energetic ions, i.e., . In that case the integration of Equation (
One major result of the model presented is that for a given laser energy , the shortest laser pulses and thus highest intensities are not necessarily optimal for TNSA. An important point is that the normalized maximum energy an ion can gain depends on the pulse duration which has a strong consequence on the maximal ion energy . This can be seen in Figure
2.2. Relativistic TNSA
The relativistic equation of motion is
Writing Equation (
In the highly relativistic regime, the ion velocity approaches the speed of light for approximately all the acceleration time, which according to Equation (
3. Radiation pressure acceleration
At the intensities available with present high-intensity lasers, it seems natural to consider RPA as a means of accelerating objects to high energy. RPA offers the most promising approach for the acceleration of plasma bunches with near-solid, or at least overcritical, density to relativistic velocities. The principle of RPA is the same as was proposed to use continuous wave lasers to drive interstellar vehicles to relativistic velocities[
The acceleration of an object with mass by the radiation pressure, where is the ion particle density, is described by Refs. [
On inspecting Equation (
In a very recent theory of laser ion acceleration from thin foils[
It is readily seen that except for a minor factor over the whole range of laser energies both optimized theories yield the same maximum ion energies. It is therefore more a question of practicability what kind of optimization one chooses, or . There might exist technical limits: it is certainly very difficult to obtain pulse durations shorter than say 1 fs. According to Equation (
In addition, experimental results are plotted in Figure
Of course, if the transverse light intensity changes, different parts of the sail will be accelerated differently changing an initially plane sail to a convex one. The equation of motion of Equation (
4. Discussion
We are very much aware that analytic descriptions of the complex laser-assisted acceleration process of ions are hampered in many aspects. In contrast, PIC simulations describe in much more detail the complex processes described here. However, we also believe that an analytic description of the multi-parameter behavior of the process and its interdependences can give a more general overlook of the strategy to obtain the required outcomes such as, e.g., maximum ion energies. At the same time we remark that also PIC simulations which in essence are based on a mean field theory may rather severely suppress microscopic interactions. We also recognize a significant overshoot of PIC simulations, promising ion beams of great quality not verified by experiment hitherto. We cite a very recent paper: ‘as pointed out recently in a number of papers circular polarized laser pulses can accelerate ions very efficiently and produce sharply peaked spectra’[
We note that such high energies as predicted by the analytical models have not been observed yet even though comparable laser conditions have been applied, for example by Mackinnon
Hence, the studies presented here are encouraging in view of future applications that rely on high-repetition-rate laser systems. For example, for medical applications such as ion tumor therapy energies exceeding 100 MeV/u are envisioned. This energy range should be attainable even with sub-100 J laser systems, while relativistic energies can be achieved with energies slightly above 100 J. In order to increase the ion energy even further, i.e., above the multi-GeV level, kJ systems such as envisioned for the Extreme Light Infrastructure will be necessary. It may also become necessary to consider novel methods not discussed here. For example, once the ions move with the speed of light, staged acceleration possibly implementing plasma wake acceleration as used presently to accelerate electrons may provide a more effective means to reach higher energies[
5. Summary
Starting from experimental results, we have shown that current theories of laser–ion acceleration can and should be optimized in order to achieve maximum ion energies. Not all theories include such a possibility, but the TNSA theory of Schreiber
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