Page 443 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 425

2
The fraction of the power transmitted into region 4 is, as before, 1−| | . The method
of impedance transformation can be applied in this manner to any number of inter-
faces. The process, although tedious, is easily handled by a computer.
The motivation for using multiple layers to reduce reflection is that the resulting
structure is less sensitive to deviations from the design wavelength if the impedances
(or refractive indices) are arranged to progressively increase or decrease from layer to
layer. For multiple layers to antireflection coat a camera lens, for example, the layer
on the lens surface would be of impedance very close to that of the glass. Subsequent
layers are given progressively higher impedances. With a large number of layers
fabricated in this way, the situation begins to approach (but never reaches) the ideal
case, in which the top layer impedance matches that of air, while the impedances
of deeper layers continuously decrease until reaching the value of the glass surface.
With this continuously varying impedance, there is no surface from which to reflect,
and so light of any wavelength is totally transmitted. Multilayer coatings designed in
this way produce excellent broadband transmission characteristics.

D12.3. A uniform plane wave in air is normally incident on a dielectric slab of
thickness λ 2 /4, and intrinsic impedance η 2 = 260
. Determine the magnitude
and phase of the reflection coefficient.

Ans. 0.356; 180 ◦

12.4 PLANE WAVE PROPAGATION
IN GENERAL DIRECTIONS
In this section, we will learn how to mathematically describe uniform plane waves
that propagate in any direction. Our motivation for doing this is our need to address
the problem of incident waves on boundaries that are not perpendicular to the prop-
agation direction. Such problems of oblique incidence generally occur, with normal
incidence being a special case. Addressing such problems requires (as always) that we
establish an appropriate coordinate system. With the boundary positioned in the x, y
plane, for example, the incident wave will propagate in a direction that could involve
all three coordinate axes, whereas with normal incidence, we were only concerned
with propagation along z.We need a mathematical formalism that will allow for the
general direction case.
Let us consider a wave that propagates in a lossless medium, with propagation
√
constant β = k = ω µ .For simplicity, we consider a two-dimensional case, where
the wave travels in a direction between the x and z axes. The first step is to consider the
propagation constant as a vector, k, indicated in Figure 12.6. The direction of k is the
propagation direction, which is the same as the direction of the Poynting vector in
2
our case. The magnitude of k is the phase shift per unit distance along that direction.

2 We assume here that the wave is in an isotropic medium, where the permittivity and permeability do
not change with field orientation. In anisotropic media (where and/or µ depend on field orientation),
the directions of the Poynting vector and k may differ.