This thesis presents results on the theoretical description of ion acceleration using ultra-short ultra-intense laser pulses. It consists of two parts. One deals with the very general and underlying description and theoretic modeling of the laser interaction with the plasma, the other part presents three approaches of optimizing the ion acceleration by target geometry improvements using the results of the first part. In the first part, a novel approach of modeling the electron average energy of an over-critical plasma that is irradiated by a few tens of femtoseconds laser pulse with relativistic intensity is introduced. The first step is the derivation of a general expression of the distribution of accelerated electrons in the laboratory time frame. As is shown, the distribution is homogeneous in the proper time of the accelerated electrons, provided they are at rest and distributed uniformly initially. The average hot electron energy can then be derived in a second step from a weighted average of the single electron energy evolution. This result is applied exemplary for the two important cases of infinite laser contrast and square laser temporal profile, and the case of an experimentally more realistic case of a laser pulse with a temporal profile sufficient to produce a preplasma profile with a scale length of a few hundred nanometers prior to the laser pulse peak. The thus derived electron temperatures are in excellent agreement with recent measurements and simulations, and in particular provide an analytic explanation for the reduced temperatures seen both in experiments and simulations compared to the widely used ponderomotive energy scaling. The implications of this new electron temperature scaling on the ion acceleration, i.e. the maximum proton energy, are then briefly studied in the frame of an isothermal 1D expansion model. Based on this model, two distinct regions of laser pulse duration are identified with respect to the maximum energy scaling. For short laser pulses, compared to a reference time, the maximum ion energy is found to scale linearly with the laser intensity for a simple flat foil, and the most important other parameter is the laser absorption efficiency. In particular the electron temperature is of minor importance. For long laser pulse durations the maximum ion energy scales only proportional to the square root of the laser peak intensity and the electron temperature has a large impact. Consequently, improvements of the ion acceleration beyond the simple flat foil target maximum energies should focus on the increase of the laser absorption in the first case and the increase of the hot electron temperature in the latter case. In the second part, exemplary geometric designs are studied by means of simulations and analytic discussions with respect to their capability for an improvement of the laser absorption efficiency and temperature increase. First, a stack of several foils spaced by a few hundred nanometers is proposed and it is shown that the laser energy absorption for short pulses and therefore the maximum proton energy can be significantly increased. Secondly, mass limited targets, i.e. thin foils with a finite lateral extension, are studied with respect to the increase of the hot electron temperature. An analytical model is provided predicting this temperature based on the lateral foil width. Finally, the important case of bent foils with attached flat top is analyzed. This target geometry resembles hollow cone targets with flat top attached to the tip, as were used in a recent experiment producing world record proton energies. The presented analysis explains the observed increase in proton energy with a new electron acceleration mechanism, the direct acceleration of surface confined electrons by the laser light. This mechanism occurs when the laser is aligned tangentially to the curved cone wall and the laser phase co-moves with the energetic electrons. The resulting electron average energy can exceed the energies from normal or oblique laser incidence by several times. Proton energies are therefore also greatly increased and show a theoretical scaling proportional to the laser intensity, even for long laser pulses.

This thesis presents results on the theoretical description of ion acceleration using ultra-short ultra-intense laser pulses. It consists of two parts. One deals with the very general and underlying description and theoretic modeling of the laser interaction with the plasma, the other part presents three approaches of optimizing the ion acceleration by target geometry improvements using the results of the first part. In the first part, a novel approach of modeling the electron average energy of an over-critical plasma that is irradiated by a few tens of femtoseconds laser pulse with relativistic intensity is introduced. The first step is the derivation of a general expression of the distribution of accelerated electrons in the laboratory time frame. As is shown, the distribution is homogeneous in the proper time of the accelerated electrons, provided they are at rest and distributed uniformly initially. The average hot electron energy can then be derived in a second step from a weighted average of the single electron energy evolution. This result is applied exemplary for the two important cases of infinite laser contrast and square laser temporal profile, and the case of an experimentally more realistic case of a laser pulse with a temporal profile sufficient to produce a preplasma profile with a scale length of a few hundred nanometers prior to the laser pulse peak. The thus derived electron temperatures are in excellent agreement with recent measurements and simulations, and in particular provide an analytic explanation for the reduced temperatures seen both in experiments and simulations compared to the widely used ponderomotive energy scaling. The implications of this new electron temperature scaling on the ion acceleration, i.e. the maximum proton energy, are then briefly studied in the frame of an isothermal 1D expansion model. Based on this model, two distinct regions of laser pulse duration are identified with respect to the maximum energy scaling. For short laser pulses, compared to a reference time, the maximum ion energy is found to scale linearly with the laser intensity for a simple flat foil, and the most important other parameter is the laser absorption efficiency. In particular the electron temperature is of minor importance. For long laser pulse durations the maximum ion energy scales only proportional to the square root of the laser peak intensity and the electron temperature has a large impact. Consequently, improvements of the ion acceleration beyond the simple flat foil target maximum energies should focus on the increase of the laser absorption in the first case and the increase of the hot electron temperature in the latter case. In the second part, exemplary geometric designs are studied by means of simulations and analytic discussions with respect to their capability for an improvement of the laser absorption efficiency and temperature increase. First, a stack of several foils spaced by a few hundred nanometers is proposed and it is shown that the laser energy absorption for short pulses and therefore the maximum proton energy can be significantly increased. Secondly, mass limited targets, i.e. thin foils with a finite lateral extension, are studied with respect to the increase of the hot electron temperature. An analytical model is provided predicting this temperature based on the lateral foil width. Finally, the important case of bent foils with attached flat top is analyzed. This target geometry resembles hollow cone targets with flat top attached to the tip, as were used in a recent experiment producing world record proton energies. The presented analysis explains the observed increase in proton energy with a new electron acceleration mechanism, the direct acceleration of surface confined electrons by the laser light. This mechanism occurs when the laser is aligned tangentially to the curved cone wall and the laser phase co-moves with the energetic electrons. The resulting electron average energy can exceed the energies from normal or oblique laser incidence by several times. Proton energies are therefore also greatly increased and show a theoretical scaling proportional to the laser intensity, even for long laser pulses.