This describes an expansion of the plasma that in the case of laser-driven ion accel-eration could lead to significant altaccel-eration in the accelaccel-eration process.
2.4 Laser-Driven Ion Acceleration 31
the polarization is transversal, the angle of incidence on the target (Θi) must be non-zero (p-polarized laser). The critical density in this case is changed by the factor ne = nccos (Θc)2. The second prerequisite is that the plasma scale length (see Eq. (2.36)) has to be large enough to create a shallow plasma density gradient. A fraction of the electric field such penetrates into the overdense region and resonantly excites electron oscillation, and thus laser energy can be coupled into the overdense plasma.
Vacuum Heating
The mechanism denominated as vacuum heating [117] is similar to resonance ab-sorption except that no resonance occurs. When a p-polarized laser impinges onto a plasma with steep density gradient and thus short plasma scale length (i.e. shorter than one wavelength), the oscillation of electrons starts and electrons are pulled from the vacuum-plasma boundary into the vacuum. Following the electric field they will turn around and be pushed back into the plasma. Due to the short scale length the laser cannot influence the electrons anymore, and thus energy has been transferred from the laser to the injected electrons.
j x B Heating
The name of ~j × B~ heating [118] is given by its origin, the second term of the Lorentz force (Eq. (2.24)) and is thus dominating for relativistic laser intensities. A longitudinal oscillation of the electrons with a frequency of 2ωLis introduced. Again analog to the effect in vacuum heating, electrons that oscillate beyond the plasma scale length are shielded and not affected by the laser anymore. In contrast to the other mechanisms, it works best for normal incidence onto the target since the force is directed along the pointing vector of the laser field. Since it requires an oscillation of electrons along one axis it can be suppressed by circularly polarized light.
2.4.2 Heating of Electrons
Since the electron heating is a mixture of different processes and dependent on many different parameters such as density and thickness of the target as well as the complete laser parameters, it is hard to be described quantitatively. Several models
[119] have developed, while the scaling law by Wilks [120] is the most common one:
kBTh ≈mec2( r
1 + a20
2 −1). (2.40)
This scaling law is directly derived from the ponderomotive potential and is thus also referred to as ponderomotive scaling. One can directly see that the hot electron temperature is dependent on the laser intensity. The laser intensity can thus be seen as one of the most important parameters for the heating of energetic electrons that affect consequently proton acceleration, as will be explained later.
The effect of a high power laser onto matter depends highly on the intensity of the laser pulse. In Fig.2.2, four different levels can be identified. At intensities below 1012 W cm=2 as typical for ns pulses, thermal heating is the dominating effect. We have described processes that lead to the ionization of matter and thus the generation of a plasma at intensities around 1014 W cm=2, generating free electrons that will cause an expansion of the plasma. Eventually with intensities beyond 1018 W cm=2 collisionless absorption mechanism are able to generate hot electrons with energies in the MeV level, that will drive the acceleration of ions.
Thermal heating ( ̴ns) v×B-force
on electrons dominates, MeV-Ion acceleration
Plasma (pre-)expansion
Plasma generation
Intensity[W/cm²]
10¹⁰ 10²²
10²⁰
10¹⁶
10¹⁴
10¹²
Time [ps]
10¹⁸
0 -5
-10 -15
-20 +5
Figure 2.2 | Plasma regimes. This graph shows relevant regimes of matter-plasma interactions at various intensities. The figure has been adapted from [16].
2.4 Laser-Driven Ion Acceleration 33
A more detailed description of this fundamental physics and scaling laws for electron energies is given by [101,104].
2.4.3 Acceleration Mechanism for Ions
We have seen several mechanism how laser energy can be coupled into a plasma and accelerate electrons to relativistic energies. We further have seen that today’s laser intensities are not sufficient to directly act on the much heavier ions and accelerate them. In all mechanisms that have been found for LION, the energy is transferred to the ions via the electrons. With the first meaningful acceleration of ions at the beginning of this century [121–123] a great number of simulations and models have been derived describing various mechanisms of laser-induced ion acceleration. The direct observation of different processes is challenging since in reality it is often a mixture of different mechanisms. Further, several of the regimes require certain and often challenging conditions such as a clean laser contrast, specific target shapes and densities or very high intensities. The most elaborated and up to this stage dominating process is the Target Normal Sheath Acceleration (TNSA) [122–124].
The name TNSA already indicates the principle of this mechanism. The accelera-tion is directed along the target normal at both the rear and the front side of the target. In the previous section it has been described how electrons can be colli-sionless heated by the laser pulse. Those heated electrons will traverse the target and eventually form an electron sheath at the rear side of the target, where conse-quently a charge separation field is formed. Electrons are pulled back to the target and simultaneously replaced by newly arriving hot electrons from the target front.
This electron sheath at the backside with a thickness of roughly the Debye length forms a quasi-static electric field acting on the ionized atoms at the rear side of the target. Consequently, due to the Coulomb forces, those ions will be accelerated. The characteristic exponential energy spectrum generated by TNSAoften characterized by its cut-off energy. Further acceleration mechanisms are:
1. Radiation pressure acceleration: The RPAmechanism has first been de-scribed in 2004 [125] as a method to accelerate ions with narrowed energy dis-tribution. The high intensity beyond 1023W cm=2required for this mechanism marks its main challenge. However, circular polarized light can circumvent this fact [126]. In 2009, the mechanism has first been demonstrated [127] and is, even though frequently sought, still challenging due to the lack in laser intensity.
2. Collisionless shock acceleration: Is similar to RPA but typically with thicker targets such that the hole boring effect of the plasma never perturbates the target completely and reaches the rear. Experimental demonstrations can be found in [128–130].
3. Coulomb explosion: The process of Coulomb explosion dominates when all electrons have been pushed out of the target leaving behind the heavier ions. Coulomb forces then cause an explosion of the purely positively charged targets in 4π [131–134], or a more directed mechanism [135], depending on geometry.
4. Magnetic vortex acceleration: This process occurring in low density tar-gets describes an acceleration based on dipole magnetic fields [136, 137].
The number of different acceleration mechanisms, that have been demonstrated in simulations or experiments is large. However, in most cases the acceleration process involves various mechanisms. In order to clearly show the effect of one mechanisms, it is mandatory to suppress other effects by an advanced choice of the laser-target-setup combination.