Page 455 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 437

EXAMPLE 12.9
Light is incident from air to glass at Brewster’s angle. Determine the incident and
transmitted angles.
Solution. Because glass has refractive index n 2 = 1.45, the incident angle will be
 
1.45
θ 1 = θ B = sin −1  n 2  = sin −1 √ = 55.4 ◦
2
2
n + n 2 2 1.45 + 1
1
The transmitted angle is found from Snell’s law, through
 

θ 2 = sin −1 n 1 sin θ B = sin −1  n 1  = 34.6 ◦
2
n 2 n + n 2 2
1
Note from this exercise that sin θ 2 = cos θ B , which means that the sum of the incident
and refracted angles at the Brewster condition is always 90 .
◦
Many of the results we have seen in this section are summarized in Figure 12.10,
in which p and s , from (69) and (71), are plotted as functions of the incident
angle, θ 1 . Curves are shown for selected values of the refractive index ratio, n 1 /n 2 .
For all plots in which n 1 /n 2 > 1, s and p achieve values of ±1at the critical angle.
At larger angles, the reflection coefficients become imaginary (and are not shown)
butnevertheless retain magnitudes of unity. The occurrence of the Brewster angle is
evident in the curves for p (Figure 12.10a) because all curves cross the θ 1 axis. This
behavior is not seen in the s functions because s is positive for all values of θ 1
when n 1 /n 2 > 1.

D12.5. In Example 12.9, calculate the reflection coefficient for s-polarized
light.

Ans. −0.355

12.7 WAVE PROPAGATION
IN DISPERSIVE MEDIA
In Chapter 11, we encountered situations in which the complex permittivity of the
medium depends on frequency. This is true in all materials through a number of pos-
sible mechanisms. One of these, mentioned earlier, is that oscillating bound charges
in a material are in fact harmonic oscillators that have resonant frequencies associated
with them (see Appendix D). When the frequency of an incoming electromagnetic
wave is at or near a bound charge resonance, the wave will induce strong oscilla-
tions; these in turn have the effect of depleting energy from the wave in its original
form. The wave thus experiences absorption, and it does so to a greater extent than
it would at a frequency that is detuned from resonance. A related effect is that the