In this chapter, everything is brought together to solve the MHD Riemann problem, the most general 1-D MHD problem one can solve semi-analytically. Non-linear waves are introduced in which the 1-D primitive equations are neither steady-state nor linearised. The fast and slow eigenkets are evaluated and their normalisation to account for the not-strictly hyperbolic nature of MHD is emphasised. A method to determine profiles of the primitive variables across slow and fast rarefaction fans is described, including Euler, switch-on, and switch-off fans. A strategy for solving the MHD Riemann problem follows, including use of a multi-variate secant root finder, sixth- order Runge–Kutta, and inverting a 5 × 5 Jacobian matrix with emphasis on characteristic degeneracy and matrix singularity. The chapter concludes with an explicit algorithm for an MHD Riemann solver including numerous examples using a solver developed by the author.

This chapter introduces the magnetic induction, B̅, to fluid dynamics. After a brief introduction establishing the ubiquity of magnetism in the universe, the ideal induction equation is derived from the idea of electromagnetic force balance and Faraday’s law. By proposing and proving the flux theorem, Alfvén’s theorem is proven to show that in an ideal MHD fluid, magnetic flux is conserved and frozen-in to the fluid. It is further shown how the introduction of Bti introduces the Lorentz force density to the momentum equation and the Poynting power density to the energy equation. Two variations of the equations of MHD are assembled, both involving the conservative variables. Finally, the vector potential, magnetic helicity, and magnetic topology are introduced in an optional section where the link to solar coronal flux loops is made.

By linearising the equations of HD developed in Chapter 1, the wave equation for the propagation of sound is derived. This is examined from two approaches: direct solution of the wave equation and examining normal modes to convert the problem to one of linear algebra. This introduces the very important concepts of eigenvalues (characteristic speeds) and eigenkets (right eigenvectors) along with the role they play in examining fluid dynamics in terms of waves. From the 1-D, non-linearised, steady-state equations, the Rankine–Hugoniot jump conditions are derived from which the conditions for tangential/contact discontinuities and shocks are developed. An optional section considers the phenomenon of bores and hydraulic jumps, while the last section introduces concepts such as streamlines and stream tubes culminating with Bernoulli’s theorem applied to an incompressible fluid, a subsonic compressible fluid, and a supersonic compressible fluid.

This chapter serves as the “practice chapter” for the main goal of Part I: solving the MHD Riemann problem. Lagrangian and Eulerian frames of reference are introduced from which the three Riemann invariants of HD are identified. Space-time diagrams are introduced as a useful visual and conceptual aid in understanding the role of characteristic paths through a continuum, which is in keeping with the text’s underlying approach of treating fluid dynamics as a form of wave mechanics. The Riemann problem for HD is defined and a method of characteristics is introduced whose main purpose is to understand qualitatively how the solution to the HD Riemann problem begins to unfold. In so doing, shocks and contact discontinuities are rediscovered and rarefaction fans are introduced. It is shown how examining the eigenkets leads to profiles of the primitive variables across a rarefaction fan which ultimately leads to a semi-analytic solution to the HD Riemann problem.

In plasmas whose density is underdense the laser pulse can propagate through the plasma, depositing energy and driving plasma waves. The diverse effects seen in plasmas of this density regime are the subject of this chapter. The interplay of field ionization of a gas target, plasma heating and subsequent effects on laser propagation is scrutinized with phenomena such as plasma-induced defocusing and filamentation the subject of chapter sections. The self-consistent response of the plasma subject to a traversing intense pulse is modeled using the quasi-static approximation, illustrating how the ultrafast laser pulse can excite plasma waves. The impact of relativistic self-focusing is assessed and its interplay with those plasma waves discussed, leading to complex propagation effects. A section then addresses instabilities in the laser plasma interaction. The final sections of the chapter discuss how the production of these plasma wave wakefields can be used to accelerate electrons, with a range of regimes described ranging from linear, to nonlinear bubble and beatwave acceleration. A concluding section discusses betatron oscillations of electrons in the bubble acceleration regime.

This chapter opens with a discussion of the definition of the strong field physics, high-intensity regime, arguing that the strong field regime is entered, when considering interactions with atoms and molecules, when the light intensity is high enough that traditional quantum perturbation theory breaks down. If considering interactions in plasmas, the light field can be considered “strong” when the laser field strength is high enough that it dominates the thermal motion of electrons in the plasma. In both cases it is argued that the strong field regime begins at light intensity near 1014 W/cm2. The chapter then goes on to recount a brief history of research in strong field high intensity laser physics, highlighting major achievements in the field since its inception with the initial pioneering publication by Keldysh on strong field atomic ionization. A historical overview of both the atomic-molecular and plasma physics aspects of the field are presented. The chapter concludes with some comments on mathematical notation employed throughout the book.

Utilizing theoretical models from the previous chapter on multiphoton, tunnelling and above-threshold ionization, this chapter presents models for the ionization and fragmentation of small molecules in strong laser fields. Ionization models from the strong field approximation and a molecular tunneling model are presented, augmented with considerations of the additional complications arising from multiatom systems such as vibrational excitation, multielectron effects and molecular alignment. Mechanisms for aligning a linear molecule in a moderate-intensity laser field are discussed, followed by a section scrutinizing the fate of molecular bonds in moderate fields. Aspects of molecular bond evolution such as bond softening and above-threshold dissociation are explored using the Floquet theory of quantum systems in strong fields. The final portion of the chapter describes the dynamics of Coulomb explosions of diatomic molecules subsequent to laser field ionization, and the critical ionization atomic separation distance at which field ionization is enhanced. A concluding section considers fragmentaion of polyatomic molecules.

In the spatial realm between quantum systems like atoms and macroscopic-scale bulk plasmas rest atomic clusters. The response of these nanoscopic-scale atomic assemblages to an intense laser field is the focus of this chapter. After a survey of how clusters can be made, the chapter illustrates how clusters in intense fields can be often described by the collective oscillations of the entire cloud of electrons in clusters of ionized atoms. The cluster behavior can be described in many regimes as a nanoscale “nanoplasma.” How a nanoplasma absorbs intense laser light through the giant dipole resonance is discussed, and this model is employed to explain a range of laser–cluster interactions. The interplay of “inner” ionization and the ejection of freed electrons from the cluster whole in “outer” ionization is explored. Collisional and collisionless absorption of energy by the nanoplasma is considered in linear and nonlinear regimes. Then the subsequent explosion, through electron thermal pressures or Coulomb explosion forces, is detailed. An attempt then is made to integrate these models into regimes of interplay among ionization, absorption and explosion of the clusters.

Before subsequent chapters delve into the physics of matter in intense focused laser fields, this chapter first surveys the laser technologies most commonly employed for experimental investigations of high-intensity laser interaction physics. Chirped pulse amplification (CPA), the pioneering invention that has enabled much of modern research on strong field physics, is explained, and the laser gain media most commonly used with the technique are surveyed. The next sections of the chapter are devoted to presenting the physics and technology behind the various components of a modern CPA laser, starting with mode-locked oscillators for seeds, pulse stretchers and compressors and power amplifier chains. Key topics behind the successful deployment of such lasers, such as pulse temporal contrast control, temporal phase control, and pulse focusing, are briefly considered. Other laser technologies used in strong field physics research are also briefly discussed in the concluding section.

Atoms subject to intense laser light will witness their bound electrons undergo nonlinear oscillatory motion and subsequent field-driven ionization. This nonlinear electron motion will reradiate electromagnatic waves at harmonic multiples of the laser frequency, often to high nonlinear orders. This chapter explores this high-order harmonic generation, considering the single-atom response combined with the coherent addition from many atoms leading to spatially coherent, short-wavelength radiation. After a phenomenological justification of high harmonics in which a quasi-classical three-step model is employed to describe harmonic generation from field-ionizing atoms, the chapter launches into a quantum description of the single-atom nonlinear dipole reponse using the strong field approximation. The following sections then address the effects of propagation and phase-matching through an extended media of nonlinearly driven atoms, assessing the effects of the geometric, intensity-dependent, and plasma-induced phase on the spatial coherence of the generated harmonics. The concluding section looks at the effect of attosecond pulse generation that accompanies harmonic production.

Intense irradiation of solid targets creates an overdense plasma surface which absorbs and reflects incident laser radiation, the subject of this final chapter. Following a survey of the physics of the plasma formation, heat transport and hydrodynamic expansion, the range of effects that absorb the laser energy are each considered, starting with collisional absorption, followed by detailed exploration of collisionless absorption mechanisms including resonance absorption, vacuum heating and JxB heating. The hot electron production and transport that accompanies these absorption mechanisms is explored, followed by assessment of ponderomotive force effects on the plasma surface, with steepening and hole-boring physics elucidated. The reradiation of high harmonic emission from the plasma surface is discussed in the context of the oscillating mirror model. The acceleration of ions from solid targets is then described with particular attention given to the target normal sheath acceleration mechanism. Why and how strong magnetic fields are produced by intense irradiation of solid-target plasmas are answered and some integrated phenomena resulting from all of this physics is surveyed.