This example builds on the rectangular photonic crystal bandstructure example . This page provides additional information and examples relevant to triangular lattice-type simulations where it is necessary to have multiple unit cells in the FDTD simulation region. All tips and methodology described in the rectangular example still apply but are not repeated here.
Overview
Understand the simulation workflow and key results
For non-rectangular lattices it is required to include multiple unit cells to create a structure that is periodic in the X, Y, and Z direction since the FDTD simulation region is always rectangular. For example, it is necessary to include two unit cells in the simulation for triangular lattices, and 4 unit cells for the FCC lattice.
Three structures are provided as examples. A 2D triangular lattice, 3D FCC lattice described in Lopez et al. [1], and 3D BCC lattice.
Run and results
Instructions for running the model and discussion of key results
- Open the FDTD file and run the associated script file.
The script will calculate and plot the bandstructure. A more detailed description of the process is presented in the rectangular lattice bandstructure example. Below is the obtained results for mentioned structures:
Important model settings
Description of important objects and settings used in this model
Multiple unit cells in the FDTD simulation region
Due to the rectangular nature of the FDTD solver region, it may be necessary to include multiple PC unit cells in the solver region.
This requires careful setup of some parameters such as sources and mesh to avoid breaking the periodicity of the system. For example, the following figure shows a PC array, where the FDTD mesh is not an integer multiple of the PC unit cell. For example, the left arrow shows a mesh line going through the center of that circle. The arrow in the middle of the figure shows a different circle where the mesh does not go through the center of the circle.
This results in each circle being meshed in a slightly different manner, which breaks the periodicity of the system and will result in incorrect results.
The parameter “spectrum” the result available from the analysis group which outputs the Fourier transform of captured time signals can be utilized to further investigate the validity of obtained peaks. It is particularly important for the mesh and sources to be set up with this issue in mind, as described below:
Dipole Sources
To avoid problems with artificial zone folding, we must have matching dipole sources in each unit cell. Each of the matching dipoles must be in exactly the same position within its unit cell. The dipole phase must also be adjusted according to the following formula:
$$ \begin{array}{l}{\Delta \theta=-\frac{180}{\pi} \vec{k} \cdot \Delta \vec{r}} \\ {\text { where }} \\ {\Delta \theta=\text { phase offset from reference dipole }} \\ {\overrightarrow{\mathrm{k}}=\text { simulation wave vector }} \\ {\Delta \overrightarrow{\mathrm{r}}=\text { position offset from reference dipole }}\end{array} $$
This is addressed by the dipole cloud analysis object.
It is also important for the dipole source to be sufficiently away from the simulation region (the grey box defining the object should not meet the simulation region). Otherwise the intended symmetry of the unit cells can be disrupted. This should be confirmed by the user before running the simulation.
Mesh
For non-rectangular lattices, the mesh should be adjusted so each unit cell included in the simulation region is meshed in exactly the same way (i.e. the mesh lines fall at the same positions relative to the structure for each unit cell). For example, in the FCC3D case the number of mesh cells in each direction should be divisible by 4 in order to ensure that each sphere in the simulation region is meshed the same way.
Updating the model with your parameters
Instructions for updating the model based on your device parameters
Common parameters to change include:
- Background refractive index: Set in the FDTD solver region.
- PC circle radius and refractive index: Set in the hex_pc object.
- Lattice constant, a: Set in the model setup script.
- Frequency range of interest: Set in the model setup script.
If you make more substantial changes, such as adding new objects, it is important to understand that some object properties are set in the model setup script. This is a convenient way to ensure the various objects have consistent settings. For example, the Bloch vector must be set in both the sources and simulation region. Ensure this parameterization is not broken when making more significant changes.
Taking the model further
Information and tips for users that want to further customize the model
Please refer to the rectangular photonic crystal bandstructure calculation example , for more information on how the simulation can be further customized.
Additional resources
Additional documentation, examples and training material
Related publications
- Lopez et al., Effective refractive index and group velocity determination of three-dimensional photonic crystals by means of white light interferometry, Physical Review B 73, 125103, 2006.