6.4 A journey through parameter space 195n^2 xbecomes complex is called a region of inaccessibility and is a region where Eq.(6.56)
does not have real roots. If a plasma is non-uniform in thexdirection so thatS,P,andD
are functions ofxandω^2 <ω^2 pe+ω^2 piso thatPis negative, the boundaries of a region of
inaccessibility (if such a region exists) are the locations where thesquare root in Eq.(6.62)
vanishes, i.e., where there is a solution forS(S−P)−D^2 =±
√
− 4 PD^2 n^2 z.6.4 A journey through parameter space
Imagine an enormous plasma where the density increases in thexdirection and the mag-
netic field points in thezdirection but increases in theydirection. Suppose further that a
radio transmitter operating at a frequencyωis connected to a hypothetical antenna which
emitsplane waves, i.e. waves with spatial dependenceexp(ik·x).These assumptions are
somewhat self-contradictory because, in order to excite plane waves, an antenna must be
infinitely long in the direction normal tokand if the antenna is infinitely long it cannot be
localized. To circumvent this objection, it is assumed that the plasma is so enormous that
the antenna at any location is sufficiently large compared to the wavelength in question to
emit waves that are nearly plane waves.
The antenna is located at some pointx,yin the plasma and the emitted plane waves
are detected by a phase-sensitive receiver. The positionx,ycorresponds to a point in CMA
space. The antenna is rotated through a sequence of anglesθand as the antenna is rotated,
an observer walks in front of the antenna staying exactly one wavelengthλ=2π/kfrom
the face of the antenna. Sinceλis proportional to 1 /n=ω/kcat fixed frequency, the
locus of the observer’s path will have the shape of a wave normal surface, i.e., a plot of 1 /n
versusθ.
Because of the way the CMA diagram was constructed, the topology of one of the two
cold plasma modes always changes when a bounding surface is traversed. Which mode is
affected and how its topology changes on crossing a bounding surface can be determined
by monitoring the polarities of the four quantitiesS,P,R,Lwithin each bounded volume.
Pchanges polarity only at theP=0bounding surface, butRandLchange polarity when
they go through zero and also when they go through infinity. Furthermore,S=(R+L)/ 2
changes sign not only whenS=0but also atR=∞and atL=∞.
A straightforward way to establish how the polarities ofS,P,R,Lchange as bounding
surfaces are crossed is to start in the extreme lower left corner of parameter space, corre-
sponding toω^2 >>ω^2 pe,ω^2 ce.This is the limit of having no plasma and no magnetic field
and so corresponds to unmagnetized vacuum. The cold plasma dispersion relation in this
limit is simplyn^2 =1;i.e., vacuum electromagnetic waves such as ordinary light waves
or radio waves. HereS=P=R=L=1because there are no plasma currents. Thus
S,P,R,Lare allpositivein this bounded volume, denoted as Region 1 in Fig.6.2 (regions
are labeled by boxed numbers). To keep track of the respective polarities, a small cross is
sketched in each of the 13 bounded volumes. The signs ofLandRare noted on the left
and right of the cross respectively, while the sign ofSis shown at the top and the sign of
Pis shown at the bottom.
In traversing from region 1 to region 2,Rpasses through zero and so reverses polar-
ity but the polarities ofL,S,P are unaffected. Going from region 2 to region 3,Spasses
through zero so the sign ofSreverses. By continuing from region to region in this manner,