Fundamentals of Plasma Physics

6.2 Dielectric tensor 191

square root in Eq.(6.49) is

√

B^2 − 4 AC=

√

(RL−SP)^2 sin^4 θ+4P^2 D^2 cos^2 θ (6.54)

and can only vanish ifPD=0andRL=PSsimultaneously.

(a) (b) (c)

(d) (e)

z z z

z z

ellipsoid dumbell wheel

Figure 6.3: (a), (b), (c) show types of wave normal surfaces;(d) and (e) show permissible
overlays of wave normal surfaces.

These theorems provide sufficient information to characterize the morphologyof wave

normal surfaces throughout all of parameter space. In particular, the theorems show that

only three types of wave normal surfaces exist. These are ellipsoid, dumbbell, and wheel

as shown in Fig.6.3(a,b,c) and each is a three-dimensional surface symmetric about thez

axis.

We now discuss the features and interrelationships of these three types of wave normal

surfaces. In this discussion, each of the two modes in Eq.(6.49) is considered separately;

i.e., either the plus or the minus sign is chosen. The convention is used that a mode is

considered to exist (i.e., has a wave normal surface) only ifn^2 > 0 for at least some range

ofθ;ifn^2 < 0 for all angles, then the mode is evanescent (i.e., non-propagating) for all

angles and is not plotted. The three types of wave normal surfaces are: