At this point, the reader should have all the necessary tools to create a useful FDTD model, in any number of dimensions, in almost any material. In the remainder of this book we cover a number of topics of general interest that have been found to be very useful in a number of FDTD applications. In this chapter, we will discuss modeling of periodic structures, modeling of physical features smaller than a grid cell size, the method known as Bodies of Revolution (BOR) for modeling cylindrical structures in 3D, and finally, the near-to-far field transformation, which is used to extrapolate the far-field radiation or scattering pattern from a confined FDTD simulation.

Modeling periodic structures

Quite a number of problems in numerical modeling involve structures that are periodic in one or more dimensions. Examples include photonic bandgap structures, which involve periodic arrays of dielectric structures; or arrays of antennas, in cases where the array is large enough that it can be analyzed as a periodic structure. These problems can be reduced to the modeling of a single period of the structure through the methods described in this section. Figure 12.1 shows a simple example of a periodic structure, consisting of rows of circular dielectric “balls” that repeat in the y-direction.

The main issue of complexity in modeling periodic structures, and which requires some ingenuity in the modeling effort, arises from the fact that future values of fields are involved in the FDTD update equations.