15.1 Introductory Remarks

In this chapter we introduce the concepts of signals and wave packets and demonstrate the intricate connection between the analytic properties of the Fourier transform of the wave packet (also called its spectral function) and the limitation of its velocity by the velocity of light.

15.2 Dispersion Relation: Normal and Anomalous Dispersion

In preceding chapters we derived and became familiar with the dispersion relation of a conducting medium, i.e.

which also defines the generalised dielectric constant ? (one should note that in general the relation is more complicated, in particular, as we saw in Chapter 9, if the frequency dependence of the conductivity ? is taken into account). We arrived at the expression (15.1) by assuming

together with the two Maxwell curl equations, and we obtained

The relation (15.1) is a general consequence if we assume that H does not vanish anywhere in space. We now want to study the dispersion relations as functions of the frequency ?.

We had also defined previously the generalised refractive index

so that

The phase velocity u _P and the group velocity v _G defined previously for real n( k) or ( ?) are given by

Solving the latter for d ?/ dk we obtain

One says, there is no dispersion if d ( ?)/ d ? = 0.

We now distinguish between two cases:

1. The case of normal dispersion defined by

   
   

   (in general ?/