Page 295 - Electromagnetics

with
k 1z η 1 k 2z η 2
Z 1 = , Z 2 = .
k 1 k 2
Note that we mayalso write
i

˜ t
˜ r
˜ ˜ i
˜ ˜ i
z
E =− E , E = T E k k 2 .
k 1 k
t
z
Let us summarize the fields in each region. For perpendicular polarization we have
i
˜ i
˜ i
E = ˆ yE e − jk ·r ,
⊥ ⊥
r
˜ ˜ i
˜ r
E = ˆ y ⊥ E e − jk ·r , (4.285)
⊥ ⊥ t
˜ t
˜ ˜ i
E = ˆ yT ⊥ E e − jk ·r ,
⊥ ⊥
and
r
˜ r
i
˜ t
˜ i
t
k × E k × E k × E
˜ i ⊥ ˜ r ⊥ ˜ t ⊥
H = , H = , H = . (4.286)
⊥ ⊥ ⊥
k 1 η 1 k 1 η 1 k 2 η 2
For parallel polarization we have
i
˜ i
k × H
i
˜ i − jk ·r
E =−η 1 e ,

k 1
r
˜ r
k × H
r
˜ r − jk ·r
E =−η 1 e ,

k 1
t
˜ t
k × H
t
˜ t − jk ·r
E =−η 2 e , (4.287)

k 2
and
˜ i
E i
˜ i
e
H = ˆ y − jk ·r ,

η 1
˜ ˜ i
E r
˜ r
e
H =−ˆ y − jk ·r ,

η 1
i
T E k k 2 t
˜ ˜ i
˜ t
H = ˆ y z e − jk ·r . (4.288)
t
η 2 k 1 k
z
The wave vectors are given by
i
i
i
i
i
k = (ˆ xβ 1 sin θ i + ˆ zτ cos γ ) − j(ˆ xα 1 sin θ i + ˆ zτ sin γ ), (4.289)
i
r
i
i
i
k = (ˆ xβ 1 sin θ i − ˆ zτ cos γ ) − j(ˆ xα 1 sin θ i − ˆ zτ sin γ ), (4.290)
t
t
t
t
t
k = (ˆ xβ 1 sin θ i + ˆ zτ cos γ ) − j(ˆ xα 1 sin θ i + ˆ zτ sin γ ). (4.291)
We see that the reflected wave must, like the incident wave, be a uniform plane wave.
We define the unsigned reflection angle θ r as the angle between the surface normal and
the direction of propagation of the reflected wavefronts (Figure 4.18). Since
i r
k · ˆ z = k 1 cos θ i =−k · ˆ z = k 1 cos θ r
and
i r
k · ˆ x = k 1 sin θ i = k · ˆ x = k 1 sin θ r
© 2001 by CRC Press LLC