The radiation pressure acceleration regime (RPA) promises fundamental advantages over TNSA such as monoenergetic ion beams and a much higher energy conversion efficiency. Concerning laser ion acceleration, RPA is the best of all approaches known so far, but is extremely demanding on the temporal contrast of the drive laser pulses, since targets with thicknesses on the order of a few nm have to stay intact until the peak of the laser pulse arrives. Since RPA is not subject of this thesis, the description is kept brief.
Similar to TNSA, the RPA mechanism is also based on the interaction of a relativistic laser pulse with a solid target, pushing electrons in the forward direction [23, 24, 25, 26]. In the case of RPA, however, the thickness of the foil target is in the nm range [27] (e.g. Diamond Like Carbon (DLC)). The skin depth is larger than the thickness of the foil. Hence, in spite of the the foil in solid state density, which usually leads to reflection, the leaser pulse can yet penetrate the target and pushes all electrons to the front leading to a shift of the electron density with respect to the ion background. This shift one hand causes strong acceleration fields for the ions on the but also keeps space charge fields low on the other hand since most electrons remain inside the ionic background. The circular polarization of the laser pulse leads to a smooth momentum transfer from the laser pulse to the electron population without heating it and subsequently causing a strong bipolar expansion of the foil, as would be the case with linear polarization.
16 2. Foundations of Laser Particle Acceleration Target Ions Electrons Laser (circ. pol.)
Figure 2.4: The Radiation Pressure Acceleration scheme (RPA) is shown schematically. An ultra-thin nm foil target is irradiated by a circularly polarized laser pulse ionizing all atoms. The electrons are pushed out of the nm-thick solid material and are accelerated. The constant energy transfer of the laser pulse to the electrons and the restoring forces of the ions lead to the acceleration of the target as a whole. [29].
The steady light pressure acting on the electrons is at an equilibrium with the restoring force of the ionic background. Consequently, the irradiated part of target is accelerated as a whole as shown in Fig. 2.4. The electron distribution and ion distribution shifted with respect to each other leads to a rotation in the longitudinal phase space during accelera- tion [28] and thus, to a compression of the phase space volume as it is also known from conventional acceleration. In a purely one dimensional scenario, all ions obtain similar energies and thus, a high efficiency for ion acceleration into a monochromatic spectrum can be achieved. In three dimensions, the acceleration is strongest in the center of the laser pulse, leading to a relatively poor overall efficiency for ions with the highest energy and best collimation. Therefore, a lot of research goes into the development of strategies that counteract this effect, such as flat-top laser intensity profiles or foils shaped to coun- terbalance the laser imprint, and hence strongly enhance the phase-space density of such beams [29]. This possibility constitutes one of the prime motivations for this thesis, since such beams will truly exploit the full potential of the beam optics developed here. High expectations for this novel scheme to efficiently deliver high-quality GeV ion beams have
2.4 Ion Acceleration 17
sparked a vigorous competition among different research groups. The first observation of RPA signatures could be achieved in this group [30].
Moreover, the new method for characterizing and tuning beam optical devices presented in this thesis allows a proper manipulation and transport which currently gives our group a distinct advantage over our competitors.
Chapter 3
Space Charge Effects
The laser electron acceleration yields particle beams with specific properties as described in the chapter before, among which the beam current density significantly differs from conventional particle acceleration. As a consequence, space charge effects play an important role, the understanding of which is a prerequisite for the realization of many experiments, such as Free-Electron-Lasers. Particularly for laser-accelerated electron beams, these effects have to be understood in detail.
Although the space charge effect is merely based on the Coulomb interaction, it is a many- particle problem which can hardly be treated analytically. Besides introducing to the numerical calculations, this chapter also aims at presenting a visual impression of the dy- namics of the intra-beam interaction of charged particles arguing with simple conserved quantities. Next, retardation effects in the context of space charge calculations are intro- duced, causing artifacts which cannot be neglected for high current electron beams. The minimization of of these artifacts is a topic of this thesis and will be introduced finally.
3.1
Numerical Calculation Methods
All numerical simulation approaches considered here are based on an ensemble of discrete
particles1 which are tracked through the simulation volume in the time domain and in a
6D phase space. This implies that every particle is characterized by its spatial position and momentum at a given point in time within the calculation frame of reference. It is thus only one 6D phase space coordinate for every particle which is used to determine the following coordinate at the next discrete time step2.
1On could consider particle densities instead of discrete particles.
2The alternative would imply taking into account the 4D trajectory rather than the 6D phase space coordinate. These 4D approaches (also sometimes called ”book keeping“) are not very common, also because computer performance presently does not allow for their treatment in an acceptable amount of time. 4D approaches are not considered here.
20 3. Space Charge Effects
Einstein postulates that the laws of physics are the same for all observers in uniform motion (frame of reference) relative to one another. Accordingly, the frame of reference, where the equation of motion for particle i is solved, is further on called the calculation frame of reference. All frames considered here move in the same longitudinal direction with different velocities as required when applying the special theory of relativity. The equation of motion is given by me d dtγi~vi = ~ F =−e(E~i+~vi×B~i) (3.1)
with the rest massme, the Lorentz factor γi, the velocity vi, the chargee, the forceF~, the
electric field E~ and the magnetic field B~.
The large number of real particles calls for grouping many electrons into simulation par- ticles, so called macro-particles, for reasons of computational efficiency. np particles form
a macro particle having np-times the charge, thus experiences np-times the force, and np-times the mass me. The particle’s dynamics
d dtγ~v =
npF~ npme
remains unchanged. This model certainly corresponds to np actual particles which are
forced on a constant distance. This constraint leads to artifacts which have to be minimized using a convergence test. From this point on, we refer to the total number of particlesnused
in a simulation rather than referring to the number np per macro particle. Convergence
can be identified using values such as the total energy (kinetic energy and field energy) of the system. The consideration of the mere vacuum expansion of high current beams
usually only requires a few thousand macro particles3. Many more macro particles are
required for example when evaluating the emittance of a beam which is focused. This scenario might require specific calculation methods, which yield appropriate scalings of the
calculation time depending on the number n of particles considered. Here, two methods
are introduced, the point-to-point interaction, where the calculation time scales liken2 and the Poisson solver, where it almost linearly scales with n.