Page 309 - Electromagnetics

where bythe convolution theorem

r
i
E (t) = R 1 (t) ∗ E (t). (4.318)
⊥
Here
i −1 ˜ i

E (t) = F E (ω)
⊥ ⊥
is the time waveform of the incident plane wave, while
−1 ˜

R 1 (t) = F R 1 (ω)
is the global time-domain reflection coefficient.
2
3
˜
To invert R 1 (ω), we use the binomial expansion (1 − x) −1 = 1 + x + x + x + ··· on
the denominator of (4.317), giving
˜ 2
˜
˜ 2
˜ 2
3
2
˜ 2

R 1 (ω) = [ 1 − P (ω)] 1 + [ 1 P (ω)] + [ 1 P (ω)] + [ 1 P (ω)] + ...
1
1
1
1
2 ˜ 2
2 ˜ 6
2
˜ 4
2
= 1 − [1 − ]P (ω) − [1 − ] 1 P (ω) − [1 − ] P (ω) − ··· . (4.319)
1 1 1 1 1 1 1
Thus we need the inverse transform of
˜ 2n
P (ω) = e − j2nk z,1 1 = e − j2nk 1 1 cos θ t .
1
Writing k 1 = ω/v 1 , where v 1 = 1/(µ 0 1 ) 1/2 is the phase velocityof the wave in region 1,
and using 1 ↔ δ(t) along with the time-shifting theorem (A.3) we have
˜ 2n
P (ω) = e − jω2nτ ↔ δ(t − 2nτ)
1
˜
where τ = 1 cos θ t /v 1 . With this the inverse transform of R 1 in (4.319) is
R 1 (t) = 1 δ(t) − (1 + 1 )(1 − 1 )δ(t − 2τ) − (1 + 1 )(1 − 1 ) 1 δ(t − 4τ) −· · ·
and thus from (4.318)
r
i
i
i
E (t) = 1 E (t) − (1 + 1 )(1 − 1 )E (t − 2τ) − (1 + 1 )(1 − 1 ) 1 E (t − 4τ) −· · · .
⊥ ⊥ ⊥
The reflected field consists of time-shifted and amplitude-scaled versions of the incident
field waveform. These terms can be interpreted as multiple reflections of the incident
wave. Consider Figure 4.23. The first term is the direct reflection from interface 1 and
thus has its amplitude multiplied by 1 . The next term represents a wave that pene-
trates the interface (and thus has its amplitude multiplied bythe transmission coefficient
1 + 1 ), propagates to and reflects from the conductor (and thus has its amplitude mul-
tiplied by −1), and then propagates back to the interface and passes through in the
opposite direction (and thus has its amplitude multiplied bythe transmission coefficient
for passage from region 1 to region 0, 1 − 1 ). The time delaybetween this wave and
the initially-reflected wave is given by 2τ, as discussed in detail below. The third term
represents a wave that penetrates the interface, reflects from the conductor, returns to
and reflects from the interface a second time, again reflects from the conductor, and
then passes through the interface in the opposite direction. Its amplitude has an ad-
ditional multiplicative factor of − 1 to account for reflection from the interface and an
additional factor of −1 to account for the second reflection from the conductor, and is
time-delayed by an additional 2τ. Subsequent terms account for additional reflections;
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