Page 299 - Electromagnetics

We find that when θ i <θ c the total field in region 1 behaves as a traveling wave
along x, but has characteristics of both a standing wave and a traveling wave along z
(Problem 4.7). The traveling-wave component is associated with a Poynting power flux,
while the standing-wave component is not. This flux is carried across the boundary
into region 2 where the transmitted field consists onlyof a traveling wave. By (4.161)
the normal component of time-average Poynting flux is continuous across the boundary,
demonstrating that the time-average power carried bythe wave into the interface from
region1 passes out through the interface into region2 (Problem 4.8).
Case 2: θ i <θ c . The wave vectors are, from (4.289)–(4.290) and (4.296),

i
k = ˆ xβ 1 sin θ i + ˆ zβ 1 cos θ i ,
r
k = ˆ xβ 1 sin θ i − ˆ zβ 1 cos θ i ,
t
k = ˆ xβ 1 sin θ i − j ˆ zα c ,
where

2
2
α c = β sin θ i − β 2 2
1
is the critical angle attenuation constant. The wave impedances are
η 1 β 2 η 2
Z 1⊥ = , Z 2⊥ = j ,
cos θ i α c
α c η 2
Z 1 = η 1 cos θ i , Z 2 =− j .
β 2
Substituting these into (4.283) and (4.284), we find that the reflection coefficients are
the complex quantities

˜ β 2 η 2 cos θ i + jη 1 α c jφ ⊥
⊥ = = e ,
β 2 η 2 cos θ i − jη 1 α c
˜ β 2 η 1 cos θ i + jη 2 α c jφ
=− = e ,
β 2 η 1 cos θ i − jη 2 α c
where

η 1 α c η 2 α c
φ ⊥ = 2 tan −1 , φ = π + 2 tan −1 .
β 2 η 2 cos θ i β 2 η 1 cos θ i
We note with interest that ρ ⊥ = ρ = 1. So the amplitudes of the reflected waves
are identical to those of the incident waves, and we call this the case of total internal
reflection. The phase of the reflected wave at the interface is changed from that of the
incident wave byan amount φ ⊥ or φ . The phase shift incurred bythe reflected wave
upon total internal reflection is called the Goos–H¨anchen shift.
In the case of total internal reflection the field in region 1 is a pure standing wave while
the field in region 2 decays exponentially in the z-direction and is evanescent (Problem
4.9). Since a standing wave transports no power, there is no Poynting flux into region 2.
We find that the evanescent wave also carries no power and thus the boundarycondition
on power flux at the interface is satisfied (Problem 4.10 ). We note that for anyincident
angle except θ i = 0 (normal incidence) the wave in region 1 does transport power in the
x-direction.

© 2001 by CRC Press LLC