In this example, we will use the eigenmode expansion (EME) solver in MODE to calculate the full transmission spectrum of the waveguide Bragg grating with an arbitrary number of periods. Note that in many cases, the center wavelength and bandwidth of the grating will give enough information about the performance of the grating, and a full transmission spectrum is not always necessary. In that case, it is recommended to use the much simpler FDTD approach shown in Bragg Grating Initial Design with FDTD.
A waveguide Bragg grating is an example of a 1D photonic bandgap structure where periodic perturbations to the straight waveguide form a wavelength specific dielectric mirror. These devices are often used as optical filters for achieving wavelength selective functions.
When simulating periodic structures using EME, only one unit cell of the geometry needs to be defined. In Bragg_EME.lms, the EME solver covers a single unit cell of the grating as shown below.
One port is set up at each end of the solver to calculate the transmission and reflection into the fundamental TE mode. Under the EME setup tab, we define 2 cell groups for the EME solver, one for the region with the larger waveguide width and one for the smaller waveguide width. Since the refractive index and geometry is uniform within each cell group, we only need to use 1 cell for each cell group. For the initial simulation, we will use ten modes for each cell group in the EME calculation. Note that symmetry is used here to reduce the number of modes required for this calculation.
To set the periodicity of the grating, we will define one periodic group under the "periodic group definition" table on the right side of the EME setup tab. The start and end cell groups are set to 1 and 2 respectively, and the number of periods is set to 500. This means that the unit cell (composed of 2 cell groups) will be propagated 500 times, and the final length of the device will be 160um.
Since EME is a frequency-domain method, in principle we would need to run one simulation for each wavelength of interest. This can be very time consuming since all the modes need to be re-calculated for each wavelength, and a large number of simulations are often required to accurately describe the full transmission spectrum. However, we can use the perturbative method of overriding the wavelength to rapidly sweep wavelength, which can be verified if desired by a full wavelength sweep.
The transmission spectrum of the waveguide Bragg grating with 500 periods is shown below, calculated using the wavelength sweep feature in the EME Analysis window with 400 points. Alternatively, the same transmission/reflection spectrum can also be obtained by using a parameter sweep to scan the wavelength directly, which is a good idea for final verification but this approach is much more time consuming than scanning the wavelength directly using our perturbative approach, and should only be used when the number of wavelengths is small.
The EME method is ideal for simulating the transmission spectrum of a finite-length waveguide Bragg grating since the full device can be challenging for FDTD-based methods due to the amount of computational time and memory required.
Note: Perturbative method
The wavelength sweep can also be done efficiently using the approach described in Waveguide Bragg Gratings and Resonators. This method is similar in spirit to the built-in wavelength sweep tool used here; however, the built-in tool is more general and does not require additional scripts to run.
X. Wang, “Silicon photonic waveguide Bragg gratings,” Ph.D. Thesis, University of British Columbia (2013).