Page 435 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 417

Solution. The 1.5 m spacing between maxima is λ/2, which implies that a wave-
length is 3.0 m, or f = 100 MHz. The first maximum at 0.75 m is thus at a distance
of λ/4 from the interface, which means that a field minimum occurs at the boundary.
Thus will be real and negative. We use (27) to write
s − 1 5 − 1 2
| |= = =
s + 1 5 + 1 3
So
2 η u − η 0
=− =
3 η u + η 0
which we solve for η u to obtain
1 377
η u = η 0 = = 75.4
5 5

12.3 WAVE REFLECTION FROM
MULTIPLE INTERFACES
So far we have treated the reflection of waves at the single boundary that occurs be-
tween semi-infinite media. In this section, we consider wave reflection from materials
that are finite in extent, such that we must consider the effect of the front and back
surfaces. Such a two-interface problem would occur, for example, for light incident
on a flat piece of glass. Additional interfaces are present if the glass is coated with
one or more layers of dielectric material for the purpose (as we will see) of reducing
reflections. Such problems in which more than one interface is involved are frequently
encountered; single-interface problems are in fact more the exception than the rule.
Consider the general situation shown in Figure 12.4, in which a uniform plane
wave propagating in the forward z direction is normally incident from the left onto
the interface between regions 1 and 2; these have intrinsic impedances η 1 and η 2 .A
third region of impedance η 3 lies beyond region 2, and so a second interface exists
between regions 2 and 3. We let the second interface location occur at z = 0, and so
all positions to the left will be described by values of z that are negative. The width
of the second region is l,so the first interface will occur at position z =−l.
When the incident wave reaches the first interface, events occur as follows: A
portion of the wave reflects, while the remainder is transmitted, to propagate toward
the second interface. There, a portion is transmitted into region 3, while the rest
reflects and returns to the first interface; there it is again partially reflected. This
reflected wave then combines with additional transmitted energy from region 1, and
the process repeats. We thus have a complicated sequence of multiple reflections
that occur within region 2, with partial transmission at each bounce. To analyze the
situation in this way would involve keeping track of a very large number of reflections;
this would be necessary when studying the transient phase of the process, where the
incident wave first encounters the interfaces.
If the incident wave is left on for all time, however, a steady-state situation is
eventually reached, in which (1) an overall fraction of the incident wave is reflected