Introduction

It is easily demonstrated that the fields associated with an electromagnetic wave propagating through a uniform dielectric medium of dielectric constant $\epsilon$ satisfy

$$\left(\frac{\epsilon}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\right) {\bfm E} = 0,$$ (622)

and

$$\nabla\wedge{\bfm E} = -\frac{\partial{\bfm B}}{\partial t}.$$ (623)

The plane wave solutions to these equations are well known:

$${\bfm E} = {\bfm E}_0\, {\rm e}^{{\rm i}\, ({\bfm k}\cdot{\bfm r}-\omega t)},$$ (624)

$${\bfm B} = {\bfm B}_0 \, {\rm e}^{{\rm i}\,({\bfm k}\cdot{\bfm r}-\omega t)},$$ (625)

where ${\bfm E}_0$ and ${\bfm B}_0$ are constant vectors, with

$$\frac{\omega^2}{k^2} = \frac{c^2}{\epsilon},$$ (626)

and

$${\bfm B}_0 = \frac{{\bfm k}\wedge{\bfm E}_0}{\omega}.$$ (627)

The phase velocity of the wave is given by

$$v = \frac{\omega}{k} = \frac{c}{n},$$ (628)

where

$$n = \sqrt{\epsilon}$$ (629)

is called the refractive index of the medium. It is clear that an electromagnetic wave propagates with a phase velocity which is slower than the velocity of light in a conventional (i.e., $\epsilon$ real and greater than unity) dielectric medium.

In some dielectric media $\epsilon$ is complex. This leads, from Eq. (4.4), to a complex wave vector ${\bfm k}$. For a wave propagating in the $x$-direction we obtain

$${\bfm E} = {\bfm E}_0\,\exp[\,{\rm i}\,({\rm Re}(k)\, x -\omega t)] \exp[-{\rm Im}(k) \,x].$$ (630)

Thus, a complex dielectric constant leads to the attenuation (or amplification) of the wave as it propagates through the medium in question.

Up to now, we have tacitly assumed that $\epsilon$ is the same for waves of all frequencies. In practice, this is not the case. In dielectric media $\epsilon$ is, in general, complex, and varies (in some cases, strongly) with the wave frequency, $\omega$. Thus, waves of different frequencies propagate through a dielectric medium with different phase velocities. This phenomenon is known as dispersion. Moreover, there may exist frequency bands in which the waves are attenuated (i.e., absorbed). All of this makes the problem of determining the behaviour of a wave packet as it propagates through a dielectric medium far from straightforward. Recall, that the solution to this problem for a wave packet traveling through a vacuum is fairly trivial. The packet propagates at the velocity $c$ without changing its shape. What is the equivalent result for the case of a dielectric medium? This is an important question, since nearly all of our information regarding the universe is obtained from the study of electromagnetic waves emitted by distant objects. All of these waves have to propagate through dispersive media (e.g., the interstellar medium, the ionosphere, the atmosphere) before reaching us. It is, therefore, vitally important that we understand which aspects of these wave signals are predominantly determined by the wave sources, and which are strongly modified by the dispersive media through which they have propagated in order to reach us.

The study of wave propagation through dispersive media was pioneered by two scientists, Arnold Sommerfeld and Léon Brillouin, during the first half of this century. In the following discussion, we shall stick as close as possible to Sommerfeld and Brillouin's original analysis.