182 Chapter 6. Cold plasma waves in a magnetized plasma
where, for clarity, the tildes have been dropped. It is now useful to introduce aspherical
coordinate system ink-space (or equivalently refractive index space) withˆzdefining the
axis andθthe polar angle. Thus, the Cartesian components of the refractive index are
related to the spherical components by
nx = nsinθ
nz = ncosθ
n^2 = n^2 x+n^2 z (6.19)and so Eq.(6.18) becomes
S−n^2 cos^2 θ −iD n^2 sinθcosθ
iD S−n^20
n^2 sinθcosθ 0 P−n^2 sin^2 θ
·
Ex
Ey
Ez
=0. (6.20)
6.2.1 Mode behavior atθ=0
Non-trivial solutions to the set of three coupled equations forEx,Ey,Ezprescribed by
Eq.(6.20) exist only if the determinant of the matrix vanishes. For arbitrary values ofθ,
this determinant is complicated. Rather than examining the arbitrary-θdeterminant im-
mediately, two simpler limiting cases will first be considered, namely the situations where
θ=0(i.e.,k‖B 0 )andθ=π/ 2 (i.e.,k⊥B 0 ).These special cases are simpler than the
general case because the off-diagonal matrix elementsn^2 sinθcosθvanish for bothθ=0
andθ=π/ 2.
Whenθ=0Eq.(6.20) becomes
S−n^2 −iD 0
iD S−n^20
00 P
·
Ex
Ey
Ez
=0. (6.21)
The determinant of this system is
[(
S−n^2) 2
−D^2
]
P=0 (6.22)
which has roots
P=0 (6.23)
and
n^2 −S=±D. (6.24)
Equation (6.24) may be rearranged in the form
n^2 =R, n^2 =L (6.25)where
R=S+D, L=S−D (6.26)
have the mnemonics “right” and “left”. The rationale behind the nomenclature“S(um)”
and “D(ifference)” now becomes apparent since