194 Chapter 6. Cold plasma waves in a magnetized plasma
For example, consider an antenna located in thex=0plane and having some specified
zdependence. When Fourier analyzed inz, such an antenna would excite a characteristic
kzspectrum. In the extreme situation of the antenna extending to infinity in thex=0plane
and having the periodic dependenceexp(ikzz), the antenna would excite just a singlekz.
Thus, the antenna-transmitter combination in this situation would imposekzandωbut
leavekxundetermined. The job of the dispersion relation would then be to determinekx.It
should be noted that antennas which are not both infinite and perfectly periodic will excite
a spectrum ofkzmodes rather than just a singlekzmode.
By writingn^2 x=n^2 sin^2 θandn^2 z=n^2 cos^2 θ,Eqs. (6.47) and (6.48) can be expressed
as a quadratic equation forn^2 x, namely
Sn^4 x−[S ̄(S+P)−D^2 ]n^2 x+P[S ̄^2 −D^2 ]=0 (6.56)where
S ̄=S−n^2 z. (6.57)
If the two roots of Eq.(6.56) are well-separated, the large root is found bybalancing the
first two terms to obtain
n^2 x≃S ̄(S+P)−D^2
S
, (large root) (6.58)or in the limit of largeP(i.e., low frequencies),
n^2 x≃SP ̄
S
, (large root). (6.59)The small root is found by balancing the last two terms of Eq.(6.56) to obtain
n^2 x≃P[S ̄^2 −D^2 ]
[S ̄(S+P)−D^2 ]
, (smallroot) (6.60)or in the limit of largeP,
n^2 x≃(n^2 z−R)(n^2 z−L)
S−n^2 z(smallroot). (6.61)Thus, any givenn^2 zalways has an associated largen^2 xmode and an associated smalln^2 x
mode. Because the phase velocity is inversely proportion to the refractive index, the root
with largen^2 xis called the slow mode and the root with smalln^2 xis called the fast mode.
Using the quadratic formula it is seen that the exact form of these two roots of Eq.(6.56)
is given by
n^2 x=S(S+P)−D^2 ±
√[
S(S−P)−D^2
] 2
+4PD^2 n^2 z
2 S. (6.62)
It is clear thatn^2 xcan become infinite only whenS=0.Situations wheren^2 xis complex
(i.e., neither pure real or pure imaginary) can occur whenPis large and negative in which
case the argument of the square root can become negative. In these cases,θalso becomes
complex and is no longer a physical angle. This shows that considering real angles between
0 <θ< 2 πdoes not account for all possible types of wave behavior. The regions where