Preamble

Once the amplitude of a wave is defined, the electric and magnetic energies in the waves are calculated in terms of this amplitude. However, the total energy in the waves cannot be identified in any simple way in general. The damping of waves is used to identify the total energy by relating the damping to the dissipative part of the response tensor in two ways. One way involves calculating the work done by the dissipative process and equating this to the energy lost by the waves. The other way involves including damping in terms of an imaginary part of the frequency (§11.4). The equivalence of the two ways of treating damping provides an explicit expression for the total energy in the waves. A semiclassical description of a distribution of waves is useful for both formal and practical purposes; the semiclassical description is based on regarding the waves as a collection of wave quanta.

The Electric and Magnetic Energies in Waves

In any physical theory, energy is defined in terms of its mechanical equivalent. In §15.2 this is achieved by calculating the work done by an arbitrary dissipative process and equating it to the energy lost by the waves. An important preliminary step is to define the amplitude of the waves and to calculate the electric energy in waves in Fourier space by using the fact that the electric energy density in coordinate space is given by ½ε0∣E∣2. The magnetic energy in waves is calculated in an analogous way.

Preamble

The response tensor for a dielectric depends on the polarizability of the individual atoms and molecules. In a “dense” isotropic medium, where “dense” is not a well-defined concept, the relation between the dielectric constant and the polarizability is given by the Lorenz–Lorentz relation. The polarizability needs to be calculated quantum mechanically, but many of the features of the response of a dielectric may be inferred from a classical model of forced oscillators.

The Polarizability of Atoms and Molecules

A simple model for the response of dielectric materials is based on assuming that the response results from induced electric dipole moments in the medium. One distinguishes between three classes of polarization on a microscopic level in different types of media. One class consists of media in which the polarization is attributed to deformation of atoms or molecules so that the mean centers of the positive and negative charges become slightly separated, implying that the atoms or molecules develop induced dipole moments. A second class of media consists of those in which the individual particles have intrinsic dipole moments. These moments are randomly oriented in the absence of an external field and become partially aligned in the presence of an external field. The responses for these two classes of media exist for static fields as well as for oscillating fields. The third class of media consists of charges that are free to move and such media become polarized in the sense that there is a net displacement between the positive and negative charges.

Preamble

“Bremsstrahlung” is used both as the generic name to describe emission due to an accelerated charged particle and as the specific name for the emission when this acceleration is due to the Coulomb field of another particle. Here we are concerned with bremsstrahlung in the latter more restrictive sense. Bremsstrahlung can result in the emission of waves in any wave mode in a plasma, but most interest is in the emission of transverse waves. Emission of Langmuir waves is of less interest because, unlike transverse waves in a plasma, Langmuir waves are generated efficiently by the Cerenkov process. The absorption process corresponding to bremsstrahlung is called collisional damping or, in some astrophysical literature, free–free absorption.

Qualitative Discussion of Bremsstrahlung

Bremsstrahlung due to Coulomb interactions between electrons and ions is an important emission process in a wide variety of plasmas. We mention only three general applications. (i) For laboratory plasmas and many space plasmas, bremsstrahlung is the basic thermal emission process at radio frequencies. Radio frequency emission results from distant electron–ion encounters in which the motion of the electron is perturbed only slightly by the Coulomb field of the ion. (ii) High-frequency photons, with an energy comparable with the initial energy of the electron, can result from a close encounter between an electron and an ion. So-called non-thermal bremsstrahlung due to energetic electrons is an important source of X-rays from a plasma.

Specific emission and absorption processes may be classified in several different ways. The presentation here is based on the nature of the extraneous current that acts as the source term for the emission. The simplest emission processes are “direct” emission in which the current is due solely to the motion of the emitting particle. The specific direct emission processes discussed here are (i) Cerenkov emission, which occurs for a particle in constant rectilinear motion at greater than the phase speed of the emitted wave, (ii) bremsstrahlung, which results from the accelerated motion of an electron due to the influence of the Coulomb field of an ion, and (iii) gyromagnetic emission, which is due to the accelerated motion of a particle in a magnetostatic field. To each of these processes there is a corresponding absorption process, and negative absorption is possible for both the Cerenkov and gyromagnetic processes. Another class of emission processes corresponds to scattering of waves by particles. The unscattered wave perturbs the orbit of the particle, and the emission due to this perturbed motion corresponds to the generation of the scattered radiation. Non-linear plasma currents need to be taken into account in treating scattering in a plasma. The non-linear currents allow another class of emission processes that are attributed to wavewave interactions, which include a variety of processes such as frequency doubling in non-linear optics and radiation from turbulent plasmas.

Preamble

Synchrotron emission is gyromagnetic emission from ultrarelativistic particles. It is important in the laboratory as a source of radiation in synchrotrons and as an energy loss mechanism for relativistic electrons confined by a magnetic field. Synchrotron radiation is of particular importance in astrophysics because it is the dominant emission mechanism for the vast majority of radioastronomical sources.

Forward Emission by Relativistic Particles

Before discussing synchrotron emission in particular, it is appropriate to discuss emission by relativistic particles in general. Important features of the emission by relativistic particles may be determined by the special theory of relativity and are only weakly dependent on the specific emission mechanism involved. Emission by a particle which is highly relativistic in the laboratory frame K may be inferred from its emission pattern in its rest frame K0. Let the instantaneous velocity v of the particle in K be along the z axis. The frame K0, as viewed from K is moving along the z axis at velocity v, as illustrated in Figure 24.1, and the frame K, as viewed from K0, is moving along the z0 axis at velocity –v.

In K0 the particle is momentarily at rest, and so its emission pattern is dipolar. For present purposes the only important point is that the emission pattern in K0 is not highly anisotropic. Let w0, k0, ψ0 describe a plane wave in K0, with ψ0 the angle between k0 and the z0 axis.

A medium with electromagnetic properties modifies an electromagnetic field imposed on it. The response of some media may be described satisfactorily in terms of induced dipole moments, and this is particularly the case for the response to static fields. The response to a fluctuating field may also be described in terms of the induced current density. This alternative description is used widely in plasma physics and is emphasized in the approach adopted here.

In Part Two the nature of electromagnetic responses is discussed in Chapter 6 and general properties of response tensors are summarized in Chapter 7. An understanding of the material in these two chapters is important in the discussion of waves in media in Part Three. The remaining three Chapters in Part Two are more of the nature of reference material. Although much of the material in these Chapters is referred to and used in the remainder of the book, a detailed understanding of this material is not essential before proceeding to Parts Three, Four and Five.

The formal theory of waves is developed by solving the wave equation. The condition for a solution to exist leads to a dispersion equation, and each specific solution of this equation is called the dispersion relation for a particular wave mode. An arbitrary wave mode is referred to as “the mode M”. The polarization vector for the mode M is defined as a unimodular vector along the direction of the electric vector found by solving the wave equation for waves in the mode M. Specific examples of wave modes are discussed for isotropic media, anisotropic crystals and cold magnetized plasmas. Transverse waves in a isotropic medium correspond to two degenerate wave modes, and the description of their polarization is discussed separately.

Introduction

In the last chapter it has been shown how photographic processing may be used to form speckle pattern correlation fringes (Section 3.3). The resolution of the recording medium used for this technique need be only relatively low compared with that required for holography since it is only necessary that the speckle pattern be resolved, and not the very fine fringes formed by the interference of object and holographic reference beams. The minimum speckle size is typically in the range 5 to 100 μm (Section 3.1) so that a standard television camera may be used to record the pattern. Thus video processing may be used to generate correlation fringes equivalent to those obtained photographically. This method is known as Electronic Speckle Pattern Interferometry or ESPI and was first demonstrated by Butters and Leendertz (1). Similar work has since been described by Biedermann et al. (2) and Løkberg et al. (3,4). The major feature of ESPI is that it enables real-time correlation fringes to be displayed directly upon a television monitor without recourse to any form of photographic processing, plate relocation etc. This comparative ease of operation allows the technique or speckle pattern correlation interferometry to be extended to considerably more complex problems of shape measurement (Chapter 5) and deformation analysis (Chapter 7).

Intensity correlation in ESPI is observed by a process of video signal subtraction or addition. In the subtraction process, the television camera video signal corresponding to the interferometer image-plane speckle pattern of the undisplaced object is stored electronically.

In preparing this second edition the authors have taken the opportunity to modify Chapter 4 and Chapter 7. The purpose of this work has been to present the theory for the optimization of ESPI in a more general and exact form (Chapter 4) and to extend considerably the range of applications discussed (Chapter 7). The latter is representative of the wide range of problems to which holographic and speckle techniques are now applied. Chapter 7 also includes a brief review of techniques for automatic fringe interpretation.