Page 314 - Electromagnetics

E 0 − jβ + z − jβ − z − jβ + z − jβ − z
= ˆ x e + e + j ˆ y −e + e
2

1 1
1
= E 0 e − j (β + +β − )z ˆ x cos (β + − β − )z + ˆ y sin (β + − β − )z
2
2 2
or
1
˜
E = E 0 e − j (β + +β − )z [ˆ x cos θ(z) + ˆ y sin θ(z)]
2
where θ(z) = (β + − β − )z/2. Because β + = β − , the velocities of the two circularly
polarized waves differ and the waves superpose to form a linearlypolarized wave with a
polarization that depends on the observation plane z-value. We maythink of the wave
as undergoing a phase shift of (β + + β − )z/2 radians as it propagates, while the direction
˜
of E rotates to an angle θ(z) = (β + − β − )z/2 as the wave propagates. Faraday rotation
can onlyoccur at frequencies where both the LHCP and RHCP waves propagate, and
therefore not within the stopband ω 0 <ω <ω 0 + ω M .
Faradayrotation is non-reciprocal. That is, if a wave that has undergone a rotation of
θ 0 radians bypropagating through a distance z 0 is made to propagate an equal distance
back in the direction from whence it came, the polarization does not return to its initial
state but rather incurs an additional rotation of θ 0 . Thus, the polarization angle of the
wave when it returns to the starting point is not zero, but 2θ 0 . This effect is employed
in a number of microwave devices including gyrators, isolators, and circulators. The
interested reader should see Collin [40], Elliott [67], or Liao [111] for details. We note
that for ω ω M we can approximate the rotation angle as
1 √
ω M ω M 1 √
θ(z) = (β + − β − )z/2 = ωz µ 0 1 + − 1 + ≈− zω M µ 0 ,
2 ω 0 − ω ω 0 + ω 2
which is independent of frequency. So it is possible to construct Faraday rotation-based
ferrite devices that maintain their properties over wide bandwidths.
It is straightforward to extend the above analysis to the case of a lossy ferrite. We
find that for typical ferrites the attenuation constant associated with µ − is small for all
frequencies, but the attenuation constant associated with µ + is large near the resonant
frequency(ω ≈ ω 0 ) [40]. See Problem 4.16.

4.11.7 Propagation of cylindrical waves

Bystudying plane waves we have gained insight into the basic behavior of frequency-
domain and time-harmonic waves. However, these solutions do not displaythe funda-
mental propertythat waves in space must diverge from their sources. To understand this
behavior we shall treat waves having cylindrical and spherical symmetries.

Uniform cylindrical waves. In § 2.10.7 we studied the temporal behavior of cylin-
drical waves in a homogeneous, lossless medium and found that theydiverge from a line
source located along the z-axis. Here we shall extend the analysis to lossy media and
investigate the behavior of the waves in the frequencydomain.
Consider a homogeneous region of space described bythe permittivity ˜ (ω), permeabil-
ity ˜µ(ω), and conductivity ˜σ(ω). We seek solutions that are invariant over a cylindrical
˜
˜
˜
˜
surface: E(r,ω) = E(ρ, ω), H(r,ω) = H(ρ, ω). Such waves are called uniform cylindrical
waves. Since the fields are z-independent we maydecompose them into TE and TM sets
as described in § 4.11.2. For TM polarization we mayinsert (4.211) into (4.212) to find
˜
1 ∂E z (ρ, ω)
˜
H φ (ρ, ω) = . (4.327)
jω ˜µ(ω) ∂ρ
© 2001 by CRC Press LLC