Page 301 - Electromagnetics

where s = jω and
2
r − sin θ i σ
D = 2 , B = 2 .
cos θ i 0 cos θ i
We can put (4.302) into a better form for inversion. We begin bysubtracting ⊥∞ , the
˜
high-frequencylimit of ⊥ . Noting that
√
1 − D
˜
lim ⊥ (ω) = ⊥∞ = √ ,
ω→∞ 1 + D
we can form
√ √ √
s − Ds + B 1 − D
˜ 0 ˜
(ω) = ⊥ (ω) − ⊥∞ = √ √ − √
⊥
s + Ds + B 1 + D
√
√ √
D s − s + B/D
= 2 √ √ √ √ .
1 + D s + D s + D/B
With a bit of algebra this becomes
√  B 
2 D s s + 2B 1
˜ 0 D
(ω) =−  1 −  − √ .
⊥ B B
D − 1 s + s s +
D−1 1 + D (D − 1) D−1
Now we can apply(C.12), (C.18), and (C.19) to obtain
0 −1 ˜ 0
(t) = F (ω) = f 1 (t) + f 2 (t) + f 3 (t) (4.303)
⊥ ⊥
where
2B Bt
f 1 (t) =− √ e − D−1 U(t),
(1 + D)(D − 1)
B 2 t B(t−x) Bx
f 2 (t) =−√ U(t) e − D−1 I dx,
D(D − 1) 2 0 2D
B Bt
f 3 (t) = √ I U(t).
D(D − 1) 2D

Here
I (x) = e −x [I 0 (x) + I 1 (x)]

where I 0 (x) and I 1 (x) are modified Bessel functions of the first kind. Setting u = Bx/2D
we can also write
√ Bt
2B D 2D Bt−2Du
f 2 (t) =− 2 U(t) e − D−1 I (u) du.
(D − 1) 0
Polynomial approximations for I (x) maybe found in Abramowitz and Stegun [?], making
0
the computation of (t) straightforward.
⊥
The complete time-domain reflection coefficient is
√
1 − D 0
⊥ (t) = √ δ(t) + (t).
⊥
1 + D

© 2001 by CRC Press LLC