Page 311 - Electromagnetics

4.11.6 Plane-wave propagation in an anisotropic ferrite medium
Several interesting properties of plane waves, such as Faradayrotation and the exis-
tence of stopbands, appear onlywhen the waves propagate through anisotropic media.
We shall studythe behavior of waves propagating in a magnetized ferrite medium, and
note that this behavior is shared bywaves propagating in a magnetized plasma, because
of the similarityin the dyadic constitutive parameters of the two media.
Consider a uniform ferrite material having scalar permittivity ˜ = and dyadic per-
meability ˜ ¯µ. We assume that the ferrite is lossless and magnetized along the z-direction.
By(4.115)– (4.117) the permeabilityof the medium is
 
jµ 2 0
µ 1
[ ˜ ¯µ(ω)] =  − jµ 2 µ 1 0 
0 0 µ 0
where

ω M ω 0 ωω M
µ 1 = µ 0 1 + 2 , µ 2 = µ 0 2 .
ω − ω 2 ω − ω 2
0 0
˜ ¯
The source-free frequency-domain wave equation can be found using (4.201) with ζ =
¯
˜ ¯
ξ = 0 and ˜ ¯ = I:

1 2
˜
¯
¯
∇· I ¯ · ∇− ω ˜ ¯µ · H = 0

¯
or, since ∇· A =∇ × A,
1 2
˜
˜
∇× ∇× H − ω ˜ ¯µ · H = 0. (4.320)

The simplest solutions to the wave equation for this anisotropic medium are TEM
plane waves that propagate along the applied dc magnetic field. We thus seek solutions
of the form
˜
˜
H(r,ω) = H 0 (ω)e − jk·r (4.321)
˜
where k = ˆ zβ and ˆ z · H 0 = 0. We can find β byenforcing (4.320). From (B.7) we find
that
˜
˜
∇× H =− jβˆ z × H 0 e − jβz .
ByAmpere’s law we have
˜
∇× H
˜ ˜
E = =−Z TE M ˆ z × H, (4.322)
jω
where
Z TE M = β/ω
is the wave impedance. Note that the wave is indeed TEM. The second curl is found to
be
˜
˜
− jβz
∇× ∇× H =− jβ∇× ˆ z × H 0 e .
After an application of (B.43) this becomes
˜
˜
˜
− jβz − jβz
∇× ∇× H =− jβ e ∇× (ˆ z × H 0 ) − (ˆ z × H 0 ) ×∇e .
© 2001 by CRC Press LLC