One-Dimensional Periodic Medium
The objective of this chapter is to establish a comprehensive and in-depth understanding of the wave propagation in a 1D periodic medium. The Bloch–Floquet theorem will be first introduced for setting up a framework for the fundamental mathematical analysis. Furthermore, a first-order approximation of a dielectric medium with sinusoidal variation of its relative dielectric constant will be employed as an example for illustrating the phase and dispersion relation of a wave supported in the periodic medium. The simple graphic method will also be introduced for predicting the physical phenomenon, such as contraflow interactions between space harmonics occurring in a 1D periodic medium. Additionally, a complex periodic medium consisting of multiple dielectric layers in a unit cell (period) will also be studied using two approaches: the Fourier-method and the modal-transmission-line method. In the former approach, the electric and magnetic fields are expressed in terms of a Fourier series expansion. In the latter, the rigorous modal solutions are directly determined by solving the transmission-line network subject to periodic boundary conditions.
3.1 Bloch–Floquet Theorem
If there is a periodicity along the x-axis, the wave function satisfies the following Bloch–Floquet condition:
This equation states that the field solutions, in an infinite ...