Page 294 - Electromagnetics

where
2
2
2
2
2
2
A = β − α − (β − α ) sin θ i , B = 2(β 2 α 2 − β 1 α 1 sin θ i ). (4.279)
2
2
1
1
Thus
2
t
t

τ = A + B 2 1/4 , γ = 1 tan −1 B . (4.280)
2 A
t
Renaming k as k 2z , we maywrite the transmitted wave vector as
z
t

k = ˆ xk 1x + ˆ zk 2z = k + jk
2 2
where
t
t
t
t

k = ˆ xβ 1 sin θ i + ˆ zτ cos γ , k =−ˆ xα 1 sin θ i − ˆ zτ sin γ .
2 2
Since the direction of propagation of the transmitted field phase fronts is perpendicular
to k , a unit vector in the direction of propagation is

2
t
ˆ xβ 1 sin θ i + ˆ zτ cos γ t
ˆ . (4.281)
2
k =
2 2 t 2 2
β sin θ i + (τ ) cos θ i
1
Similarly, a unit vector perpendicular to planar surfaces of constant amplitude is given
by
t
ˆ xα 1 sin θ i + ˆ zτ sin γ t
ˆ . (4.282)
2
k =
2
2
2
t 2
α sin θ i + (τ ) sin γ t
1
In general k is not aligned with k and thus the transmitted field is a nonuniform plane
ˆ
ˆ
wave.
With these definitions of k 1x , k 1z , k 2z , equations (4.274) and (4.275) can be solved si-
multaneouslyand we have
˜ t
˜ ˜ i
˜ ˜ i
˜ r
E = ⊥ E , E = T ⊥ E ,
⊥ ⊥ ⊥ ⊥
where
˜ Z 2⊥ − Z 1⊥ ˜ ˜ 2Z 2⊥
⊥ = , T ⊥ = 1 + ⊥ = , (4.283)
Z 2⊥ + Z 1⊥ Z 2⊥ + Z 1⊥
with
k 1 η 1 k 2 η 2
Z 1⊥ = , Z 2⊥ = .
k 1z k 2z
˜
Here is a frequency-dependent reflection coefficient that relates the tangential compo-
˜
nents of the incident and reflected electric fields, and T is a frequency-dependent trans-
mission coefficient that relates the tangential components of the incident and transmitted
electric fields. These coefficients are also called the Fresnel coefficients.
For the case of parallel polarization we solve (4.276) and (4.277) to find
˜ r
˜ r
t
r ˜ r
E k E E E ˜ t (k /k 2 )E ˜ t
,x x ˜ ,x z ˜
= =− = , = = T .
˜ i
˜ i
i
i
E k ˜ i E E ˜ i (k /k 1 )E ˜ i
E
,x x ,x z
Here
˜ Z 2 − Z 1 ˜ ˜ 2Z 2
= , T = 1 + = , (4.284)
Z 2 + Z 1 Z 2 + Z 1
© 2001 by CRC Press LLC