Page 290 - Electromagnetics

Since we still have k · E = 0, we mayuse the rest of (B.7) to write
ˇ
ˇ
∗ ˇ
∗ ˇ
ˇ ∗
∗
ˇ ∗
E × (k × E) = k (E · E ) + E(k · E) = k (E · E ).
ˇ ∗
ˇ ∗
ˇ
ˇ ∗
Substituting this into (4.266), and noting that E × E is purelyimaginary, we find
1 1
∗ ˇ 2
ˇ
ˇ ∗
∗
S av = Re jk × Im E × E + k |E| . (4.267)
2 ˇ ωµ ∗
Thus the vector direction of S av is not generallyin the direction of propagation of the
plane wavefronts.
Let us examine the special case of nonuniform plane waves propagating in a lossless
material. It is intriguing that k maybe complex when k is real, and the implication is
important for the plane-wave expansion of complicated fields in free space. By(4.218),
real k requires that if k = 0 then

k · k = 0.
Thus, for a nonuniform plane wave in a lossless material the surfaces of constant phase
and the surfaces of constant amplitude are orthogonal. To specialize the time-average
power to the lossless case we note that µ is purelyreal and that

E × E = (E 0 × E )e 2k ·r .
∗
∗
0
Then (4.267) becomes
1 2k ·r
ˇ 2

S av = e Re j(k − jk ) × Im E 0 × E ∗ 0 + (k − jk )|E|
2 ˇωµ
or
1 2k ·r
ˇ 2

S av = e k × Im E 0 × E ∗ 0 + k E| .
2 ˇωµ
We see that in a lossless medium the direction of energypropagation is perpendicular
to the surfaces of constant amplitude (since k · S av = 0), but the direction of energy

propagation is not generallyin the direction of propagation of the phase planes.
We shall encounter nonuniform plane waves when we studythe reflection and refrac-
tion of a plane wave from a planar interface in the next section. We shall also find in
§ 4.13 that nonuniform plane waves are a necessaryconstituent of the angular spectrum
representation of an arbitrarywave field.
4.11.5 Plane waves in layered media
A useful canonical problem in wave propagation involves the reflection of plane waves
byplanar interfaces between differing material regions. This has manydirect applica-
tions, from the design of optical coatings and microwave absorbers to the probing of
underground oil-bearing rock layers. We shall begin by studying the reflection of a plane
wave at a single interface and then extend the results to anynumber of material layers.

Reflection of a uniform plane wave at a planar material interface. Consider
two lossymedia separated bythe z = 0 plane as shown in Figure 4.18. The media are as-
sumed to be isotropic and homogeneous with permeability ˜µ(ω) and complex permittivity
c
c
˜ (ω). Both ˜µ and ˜ maybe complex numbers describing magnetic and dielectric loss,

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