Page 305 - Electromagnetics

Figure 4.21: Wave flow diagram showing interaction of incident and reflected waves for
region n.

Defining
˜ Z ⊥n − Z ⊥n−1
n = (4.308)
Z ⊥n + Z ⊥n−1
as the interfacial reflection coefficient for interface n (i.e., the reflection coefficient as-
suming a single interface as in (4.283)), and

˜ 2Z ⊥n ˜
T n = = 1 + n
Z ⊥n + Z ⊥n−1
as the interfacial transmission coefficient for interface n, we can write (4.307) as

˜
˜ ˜
˜
˜
a n+1 = a n T n P n + b n+1 P n (− n )P n .
Finally, if we define the global reflection coefficient R n for region n as the ratio of the
amplitudes of the reflected and incident waves,
˜
R n = b n /a n ,
we can write
˜
˜
˜
˜ ˜
˜
a n+1 = a n T n P n + a n+1 R n+1 P n (− n )P n . (4.309)
For n = N we merelyset R N+1 = 0 to find
˜ ˜
a N+1 = a N T N P N . (4.310)
If we choose to eliminate a n+1 from (4.305) and (4.306) we find that
˜
˜
˜
˜
b n = a n n + R n+1 P n (1 − n )a n+1 . (4.311)
For n = N this reduces to
˜
b N = a N N . (4.312)
Equations (4.309) and (4.311) have nice physical interpretations. Consider Figure 4.21,
which shows the wave amplitudes for region n. We maythink of the wave incident on

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