Page 439 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 421

the results on reflection and transmission. Equivalently, we have a single-interface
problem involving η 1 and η 3 .Now, with η 3 = η 1 ,wehavea matched input impedance,
and there is no net reflected wave. This method of choosing the region 2 thickness is
known as half-wave matching. Its applications include, for example, antenna housings
on airplanes known as radomes, which form a part of the fuselage. The antenna,
inside the aircraft, can transmit and receive through this layer, which can be shaped to
enable good aerodynamic characteristics. Note that the half-wave matching condition
no longer applies as we deviate from the wavelength that satisfies it. When this is
done, the device reflectivity increases (with increased wavelength deviation), so it
ultimately acts as a bandpass filter.
Often, it is convenient to express the dielectric constant of the medium through
the refractive index (or just index), n, defined as

√
n = r (38)
Characterizing materials by their refractive indices is primarily done at optical fre-
quencies (on the order of 10 14 Hz), whereas at much lower frequencies, a dielectric
constant is traditionally specified. Since r is complex in lossy media, the index will
also be complex. Rather than complicate the situation in this way, we will restrict
our use of the refractive index to cases involving lossless media, having = 0, and

r
µ r = 1. Under lossless conditions, we may write the plane wave phase constant and
the material intrinsic impedance in terms of the index through

√ √ nω
β = k = ω µ 0 0 r = (39)
c
and

1 µ 0 η 0
η = √ = (40)
r 0 n

Finally, the phase velocity and wavelength in a material of index n are

c
ν p = (41)
n
and

ν p λ 0
λ = = (42)
f n
where λ 0 is the wavelength in free space. It is obviously important not to confuse
the index n with the similar-appearing Greek η (intrinsic impedance), which has an
entirely different meaning.
Another application, typically seen in optics, is the Fabry-Perot interferometer.
This, in its simplest form, consists of a single block of glass or other transparent