Page 450 - Engineering Electromagnetics, 8th Edition

432 ENGINEERING ELECTROMAGNETICS

are defined through

η 1p = η 1 cos θ 1 (67)
and

η 2p = η 2 cos θ 2 (68)

Using this representation, Eqs. (65) and (66) are now in a form that enables them
to be solved together for the ratios E /E + and E 20 /E . Performing analogous
−
+
10
10
10
procedures to those used in solving (7) and (8), we find the reflection and transmission
coefficients:
E − η 2p − η 1p
p = 10 = (69)
E 10 η 2p + η 1p
+

E 20 2η 2p cos θ 1
τ p = = (70)
E 10 η 2p + η 1p cos θ 2
+
Asimilarprocedurecanbecarriedoutfors-polarization,referringtoFigure12.7b.
The details are left as an exercise; the results are
E − η 2s − η 1s
y10
s = = (71)
E + η 2s + η 1s
y10

E y20 2η 2s
τ s = = (72)
E + η 2s + η 1s
y10
where the effective impedances for s-polarization are

η 1s = η 1 sec θ 1 (73)
and

η 2s = η 2 sec θ 2 (74)

Equations (67) through (74) are what we need to calculate wave reflection and trans-
mission for either polarization, and at any incident angle.

EXAMPLE 12.7
A uniform plane wave is incident from air onto glass at an angle from the normal of
30 . Determine the fraction of the incident power that is reflected and transmitted for
◦
(a) p-polarization and (b) s-polarization. Glass has refractive index n 2 = 1.45.