In the previous 2D-periodic photonic crystal bandstructure examples, we have only considered modes that propagate within the plane of periodicity, where kx and ky vary and kz=0. This example demonstrates how to get bandstructure plots for out-of-plane propagation where kz is no longer 0.
In this example, we want to simulate the out-of-plane bandstructure for a triangular lattice with rods that are uniform and infinite in the z-direction. By using a 3D simulation with Bloch boundary conditions in the x, y, and z directions, this allows us to control the values of kz as well as kx and ky. The simulation region only needs to be one mesh cell thick in the z-direction.
The model analysis group is used to set the value of kz. In order to get the band frequencies over a range of kz values, we set up a parameter sweep task to sweep the value of kz over the desired range. In this example we are interested in the out-of-plane bandstructure at point gamma where kx=0 and ky=0, and we sweep kz from 0 to 2 in bandstructure units (SI*a/(2*pi)). The values of kx and ky in the sweep can be changed in order to get the out-of-plane bandstructure at a different point on the Brillouin zone.
Running the simulation
Open the simulation file tri_2D_kz.fsp and run the associated script file to run the parameter sweep and plot the band diagram.
The resulting band diagram closely matches the first few bands of a structure of the same material and radius/lattice constant ratio plotted in Figure 11 in Chapter 5 of Joannopoulos' text.
Plot of fs vs k
Extracted band diagram
Joannopoulos, J.D. et al. Photonic Crystals: Molding the Flow of Light. 2nd Ed. Princeton: Princeton University Press, 2008. 75-8.