αj θ¼θ c ¼ 0:
Since 1 À n
2
> 0 (i. e., n < 1) and in the total reflection region sin
2
θ À n
2
> 0, the
imaginary part of R
⊥
E is negative for any θ (i.e., 0 to π/2). On the other hand, the real
part of R
⊥
E varies from 1 to À1, as is evidenced from (8.118) and (8.119). At θ 0 that
satisfies a following condition:
sin θ 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ n 2
2
r
,
ð8:121Þ
the real part is zero. Thus, the phase shift α varies from 0 to Àπ as indicated in
Fig. 8.8. Comparing (8.121) with (8.116) and taking into account n < 1, we have
θ c < θ 0 < π=2:
Similarly, we estimate the phase change for a TM wave. Rewriting (8.112), we
have
R
k
E ¼
Àn
4 cos
2
θ þ sin
2
θ À n
2
þ 2in
2 cos θ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin
2
θ À n 2
p
n 4 cos 2 θ þ sin
2
θ À n 2
¼
Àn
4 cos
2
θ þ sin
2
θ À n
2
þ 2in
2 cos θ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin
2
θ À n 2
p
1 À n 2
ð
Þ sin
2
θ À n 2 cos 2 θ
À
Á
:
ð8:122Þ
Then, we have
R
k
E
θ¼θ c
¼ À1:
ð8:123Þ
Also at θ ¼ π/2 (i.e., grazing incidence) we have
1
i
α
=
= sin
1 +
2
= 2
0
Fig. 8.8 Phase shift α
defined in a complex plane
for the total reflection of TE
wave. The number n denotes
a relative refractive index of
D2 relative to D1. At a
critical angle θ c , α ¼ 0
318
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media