Motivational Overview

The motivating theme of these notes is the direct interaction of ultra-intense lasers with matter. Indeed, during the past decade a confluence of advances in laser science has opened the door to the study of laser-matter interaction as the new frontier of the 21st century. The extraordinarily high intensity (petawatt) of the laser pulses have pushed the relativistic, and hence nonlinear, nature of laser-matter interactions to the forefront of science. These processes are characterized by ultra-relativistic velocities and accelerations of such extreme violence that relativistic physics and mathematics is not merely optional but mandatory.

At present there is a considerable amount of highly visible experimental activity whose purpose is to grasp the properties and the nature of these extreme laser processes. However, a mathematically solid understanding in the form of principles, formulations, equations, etc., is sorely lacking. This lack extends from the laser-driven mechanics of particles, to that of plasmas, and onto that of fluids.

Our present focus is on the mechanics of particles. They are the fundamental building blocks of matter, and as such their interactions with ultra-intense laser radiation plays a fundamental role in physics. The nature of these interactions, which manifests itself through the mechanical trajectories of test particles, is controlled by the externally given laser-radiation field.

The set of all posible test particles, each with a given charge and mass, placed into such a field, and hence subject to well-determined measurable dynamical motions, forms a dynamical system. Thus a dynamical system is identified uniquely by the given laser radiation field.

There are a number of dynamical systems whose importance derives from laser fields readily implemented in the laboratory.

1. Particles moving in the e.m. field of a travelling wave.
2. Particles moving in the e.m. field of a standing wave.
3. Particles moving in the e.m. field of two counter-propagating waves, one having finite but small amplitude compared to the other,
4. Particles moving in the bichromatic e.m. field of two laser beams: one ultra-intense beam with a second weaker counter-propagating beam at a harmonic frequency.
5. Particles moving in the e.m. field of a plane wave beam of finite width.
6. Particles moving in the e.m. field of a weakly focussed travelling (or standing) wave Gaussian beam.

The first four, depicted in Figure 1, are charaterized by flat plane wave phase fronts infinite in extent. The last one has weakly curved phase fronts of finite extent.

Figure 1: Amplitude graphs of four e.m. wave fields. Their potentials charterize four distinct laser-driven particle systems.

Even though Figures 1(a) and 1(c) are depict two archetypical laser fields, it is Figure 1(b) which is usually achieved in the laboratory. However, as shown in Figure 2, the laser field of two counter-propagating waves is equivalent to the superposition of a standing wave and a travelling wave,

$$A\sin\omega (t+z) +B\sin\omega (t-z)=2A\sin\omega t \cos\omega z+(B-A)sin\omega (t-z)\, ~.$$

Figure 2: Two counter-propagating waves of equal frequency are equivalent to a superposition of a standing wave and a travelling wave.

Thus a particle moving in the e.m. field of Figure 1(b) is a combination of the motions in the two archetypical laser fields. Consequently, a mathematical formulation of this combined motion demands a mathematical formulation of the laser-driven particle mechanics for each of these two fields.