Page 308 - Electromagnetics

Figure 4.22: Interaction of a uniform plane wave with a conductor-backed dielectric slab.

them. This technique finds use in optical coatings for lenses and for reducing the radar
reflectivityof objects.
As a second example, consider a lossless dielectric slab with ˜ = 1 = 1r 0 , and ˜µ = µ 0 ,
backed by a perfect conductor and immersed in free space as shown in Figure 4.22. A
perpendicularlypolarized uniform plane wave is incident on the slab from free space
and we wish to find the temporal response of the reflected wave byfirst calculating the
frequency-domain reflected field. Since 0 < 1 , total internal reflection cannot occur.
Thus the wave vectors in region 1 have real components and can be written as
i
r
k = k x,1 ˆ x + k z,1 ˆ z, k = k x,1 ˆ x − k z,1 ˆ z.
1 1
From Snell’s law of refraction we know that
k x,1 = k 0 sin θ i = k 1 sin θ t
and so
ω 2

2
2
k z,1 = k − k =
1 x,1 1r − sin θ i = k 1 cos θ t
c
where θ t is the transmission angle in region 1. Since region 2 is a perfect conductor we
˜
have R 2 =−1. By(4.315) we have
˜ 2
1 − P (ω)
˜ 1
R 1 (ω) = , (4.317)
˜ 2
1 − 1 P (ω)
1
where from (4.308)
Z 1 − Z 0
1 =
Z 1 + Z 0
is not a function of frequency. By the approach we used to obtain (4.300) we write
r
˜
˜ i
˜ r
E (r,ω) = ˆ yR 1 (ω)E (ω)e − jk (ω)·r .
1
⊥ ⊥
So

ˆ r
k · r
r
1
r
E (r, t) = ˆ yE t −
⊥
c
© 2001 by CRC Press LLC