Structures that are periodic in two dimensions are a common PBG application. In such structures, light is confined in the third dimension by traditional index guiding. Typical examples include waveguide slabs with a periodic set of holes (PCS), or a lattice of rods of finite vertical extent. These structures are treated similarly in BandSOLVE. These structures have attracted considerable interest since they possess many of the features of full 3D photonic crystals, but are substantially easier to manufacture.
Creating the Structure
This example consists of a cubic array of dielectric rods with refractive index of sqrt(12) in the XZ plane in a background material of air as discussed in Ref. .
It is very important to appreciate that a PCS is a 3D problem; even though the structure is only periodic in two dimensions, the index distribution still varies in the third dimension because the slab has finite height. Both the slab and the surrounding substrate must be included in the numerical domain if the correct answer is to be obtained.
The simulation results for this structure are:
This result shows the use of a light line, which is necessary to filter out radiation modes, and the consideration of parity, which allows modes to be classified as either even or odd. In this case, there is a clear bandgap for odd modes between 0.34 and 0.42 whereas in the even case we find that no such bandgap exists. This means that there is a quasi-TE bandgap for this structure.
 S. Johnson, S. Fan, R. Villeneuve, and J.D. Joannopoulos, Guided modes in Photonic Crystal Slabs, Phys. Rev. B 60, 5751 (1999).