Perpendicular EM Waves in Magnetized Plasmas
Consider electromagnetic wave propagation, at arbitrary frequencies, perpendicular
to the equilibrium magnetic field in a magnetized plasma. When
, the eigenmode equation
(9.168) simplifies to

(9.192) 
One obvious way of solving this equation is to have
, or

(9.193) 
with the eigenvector
. Because the wavevector now points in the
direction, this is clearly a transverse wave polarized with its electric
field parallel to the equilibrium magnetic field. Particle motions are
along the magnetic field, so the mode dynamics are completely unaffected
by this field. Thus, the wave is identical to the
electromagnetic plasma wave found
previously in an unmagnetized plasma. (See Section 9.3.) This wave is known as the ordinary,
or , mode.
The other solution to Equation (9.192) is obtained by setting the determinant
involving the  and components of the electric field to zero. With the help of
the identity
, the
dispersion relation reduces to

(9.194) 
with the associated eigenvector
.
Let us, first of all, search for the cutoff frequencies, at which goes to
zero.
According to Equation (9.194), these frequencies are the roots of and .
In fact, we have already solved these equations;
there are two cutoff frequencies, and ,
that are specified by Equations (9.188) and (9.191), respectively.
Let us, next, search for the resonant frequencies, at which goes to
infinity. According to Equation (9.194), the
resonant frequencies are solutions of

(9.195) 
The roots of this equation can be obtained as follows. First, we note that
if the first two terms in the middle are equated to zero then we obtain
,
where

(9.196) 
If this frequency is substituted into the third term in the middle then the result is
far less than unity. We conclude that
is a good approximation
of one of the roots of Equation (9.195). To
obtain the second root, we make use of the fact that the product of the square
of the roots
is

(9.197) 
We, thus, obtain
, where

(9.198) 
The first resonant frequency,
, is greater than the
electron cyclotron or plasma frequencies, and is called the upper hybrid
frequency. The second resonant frequency,
, lies between the
electron and ion cyclotron frequencies, and is called the
lower hybrid frequency.
Unfortunately, there is no simple explanation of the origins of the
two hybrid resonances in terms of the motions of individual particles.
At low frequencies, the mode in question
reverts to the compressionalAlfvén wave
discussed previously. Note that the shearAlfvén wave does not
propagate perpendicular to the magnetic field.
Figure 9.4:
Schematic diagram showing the dispersion relation for an electromagnetic wave propagating perpendicular to the
magnetic field in a magnetized plasma.

Using the previous information, and the easily demonstrated fact that

(9.199) 
we deduce that the dispersion curve for the mode in question
takes the form sketched in Figure 9.4.
The lowest frequency branch corresponds to the compressionalAlfvén wave.
The other two branches constitute the extraordinary, or , mode.
The upper branch is basically a linearly polarized (in the direction)
electromagnetic wave, somewhat modified by the presence of the plasma. This
branch corresponds to a wave that
propagates in the absence of an equilibrium magnetic field. The lowest
branch corresponds to a wave that does not propagate in the absence of an
equilibrium field. Finally, the middle branch corresponds to a wave that
converts into an electrostatic plasma wave in the absence of an equilibrium
magnetic field.
Wave propagation at oblique angles is generally more complicated than
propagation parallel or perpendicular to the equilibrium magnetic field,
but does not involve any new physical effects (Stix 1992; Swanson 2003).