Wave Propagation in Multiple Dielectric Layers
As is well known from the textbooks in electromagnetics, the plane-wave solutions in a uniform medium can be decomposed into two polarized modes. Therefore, each one can be analyzed with relative ease. However, in the presence of a discontinuity between periodic structures and a uniform medium, the single polarization mode can no longer satisfy the electromagnetic boundary condition when the plane wave is incident obliquely (out-of-plane propagation). The superposition of the two polarized modes is needed to determine the general solution. Consequently, our main effort in this section is to introduce a rigorous and systematic way for putting the two polarization modes back together. By invoking the mathematics of dyadic and eigenvalue problems, we will prove that the two orthogonal eigenbases correspond to TE and TM waves and the two associated eigenvalues relate to the wave (or characteristic) impedances of the two polarizations. Moreover, the wave propagating along the longitudinal direction can be interpreted using the commonly used transmission-line equation. Namely, through the rigorous mathematical process described in this chapter, the electromagnetic field problem in a uniform layer can be transformed into the electric circuit analogy using the transmission-line representation.
2.1 Plane-Wave Solutions in a Uniform Dielectric Medium
We begin with the Maxwell equations in the frequency domain, where μ and are the ...