TABLE OF CONTENTS For an infinite structure, it is impossible to directly simulate it using limited computation resources. Thus, the computation domain needs to be truncated using proper boundary conditions. The perfectly matched layers (PML) discussed in the previous section are used to absorb radiating energy from antennas or scatterers that have finite sizes. In many applications, especially in artificial electromagnetic materials, an EM structure itself extends to infinity in a periodic manner. Typical examples include corrugated waveguides, frequency selective surfaces, electromagnetic band gap structures, and double negative materials. In this situation, the PML that absorbs radiating waves are no longer applicable and a new boundary condition needs to be developed to truncate the computational domain. For this purpose, a periodic boundary condition (PBC) that models the effect of periodic replication is introduced to truncate the computational domain so that only a single unit cell needs to be simulated [16].
All periodic boundary conditions are developed based on the Floquet theory [17]. For a periodic structure with periodicity p along the x direction, electromagnetic fields at two boundaries x = 0 and x = p satisfy the following equations in the frequency domain:
The exponential term represents a propagation phase delay, which is determined by the propagation constant k x and the periodicity p.
For a waveguide problem, the propagation constant k x can be obtained from the dispersion relation. However, the dispersion relation for a...
