Page 353 - Electromagnetics

Figure 4.34: Geometry of a periodic stratified medium with each cell consisting of two
material layers.

4.14.2 Examples of periodic systems

Plane-wave propagation within a periodically stratified medium. As an exam-
ple of wave propagation in a periodic structure, let us consider a plane wave propagating
within a layered medium consisting of two material layers repeated periodically as shown
in Figure 4.34. Each section of two layers is a cell within the periodic medium, and we
seek an expression for the propagation constant within the cells, κ.
We developed the necessary tools for studying plane waves within an arbitrary layered
medium in § 4.11.5, and can apply them to the case of a periodic medium. In equations
(4.305) and (4.306) we have expressions for the wave amplitudes in any region in terms
of the amplitudes in the region immediately preceding it. We may write these in matrix
form by eliminating one of the variables a n or b n from each equation:

(n) (n)
T T
11 12 a n+1 = a n (4.423)
T (n) T (n) b n+1 b n
21 22
where
˜ −1
T (n) = 1 Z n + Z n−1 P ,
11 n
2 Z n
(n) 1 Z n − Z n−1
˜
T = P n ,
12
2 Z n
(n) 1 Z n − Z n−1
˜ −1
T = P ,
21 n
2 Z n
˜
T (n) = 1 Z n + Z n−1 P n .
22
2 Z n
Here Z n represents Z n⊥ for perpendicular polarization and Z n for parallel polariza-
tion. The matrix entries are often called transmission parameters, and are similar to
the parameters used to describe microwave networks, except that in network theory the
wave amplitudes are often normalized using the wave impedances.We may use these

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