Page 452 - Engineering Electromagnetics, 8th Edition

434 ENGINEERING ELECTROMAGNETICS

12.6 TOTAL REFLECTION AND TOTAL
TRANSMISSION OF OBLIQUELY
INCIDENT WAVES

Now that we have methods available to us for solving problems involving oblique in-
cidence reflection and transmission, we can explore the special cases of total reflection
and total transmission.We look for special combinations of media, incidence angles,
and polarizations that produce these properties. To begin, we identify the necessary
2
condition for total reflection. We want total power reflection, so that | | = = 1,
∗
where is either p or s . The fact that this condition involves the possibility of a
complex allows some flexibility. For the incident medium, we note that η 1p and
η 1s will always be real and positive. On the other hand, when we consider the second
medium, η 2p and η 2s involve factors of cos θ 2 or 1/ cos θ 2 , where

1/2
2
2 2
1/2 n 1
cos θ 2 = 1 − sin θ 2 = 1 − sin θ 1 (75)
n 2
where Snell’s law has been used. We observe that cos θ 2 , and hence η 2p and η 2s ,
become imaginary whenever sin θ 1 > n 2 /n 1 . Let us consider parallel polarization,
for example. Under conditions of imaginary η 2p , (69) becomes

j|η 2p |− η 1p η 1p − j|η 2p | Z
p = =− =−
j|η 2p |+ η 1p η 1p + j|η 2p | Z ∗
where Z = η 1p − j|η 2p |. We can therefore see that p = 1, meaning total power
∗
p
reflection, whenever η 2p is imaginary. The same will be true whenever η 2p is zero,
which will occur when sin θ 1 = n 2 /n 1 .We thus have our condition for total reflection,
which is

n 2
sin θ 1 ≥ (76)
n 1
From this condition arises the critical angle of total reflection, θ c , defined through

n 2
sin θ c = (77)
n 1
The total reflection condition can thus be more succinctly written as

(for total reflection) (78)
θ 1 ≥ θ c
Note that for (76) and (77) to make sense, it must be true that n 2 < n 1 ,or the wave
must be incident from a medium of higher refractive index than that of the medium
beyond the boundary. For this reason, the total reflection condition is sometimes
called total internal reflection; it is often seen (and applied) in optical devices such