In this example, we will use 3D FDTD simulations to access how the performance of the Bragg grating is affected by geometric parameters such as the corrugation depth and misalignment.

## Background

A waveguide Bragg grating is an example of a 1D photonic bandgap structure where periodic perturbations to the straight waveguide forms a wavelength specific dielectric mirror. These devices are often used as optical filters for achieving wavelength selective functions.

## Simulation Setup

For this example, we will use a 3D FDTD simulation of a single unit cell of the grating to find the center wavelength and bandwidth of the infinitely periodic device. In Bragg_FDTD_unit_cell.fsp, the simulation region contains exactly one unit cell of the grating. Bloch boundaries are used for the x min/max boundaries, which allows us to set the kx value for the infinitely periodic device. A mode source is used as the excitation, and the bandstructure analysis group is used to calculate the spectrum. For this simulation, we are interested in the spectrum at the band edge k_{x} = π/a, which will give us the size and location of the band gap of the grating.

## Simulation Results

### Bandwidth and center wavelength

Once the simulation finishes running, the spectrum at k_{x} = π/a is returned by the bandstructure analysis group, as shown below. From the two peaks in the spectrum we can find the size and location of the band gap, which correspond to the bandwidth and center wavelength of the Bragg grating. Note that the width of the peaks decreases with increasing simulation time; therefore, the simulation time should be long enough to generate clear separate peaks. The original simulation time in the attached file Bragg_FDTD_unit_cell.fsp is 1250fs.

Spectrum results for different simulation times.

### Coupling coefficient as a function of grating depth

Even though this FDTD method applies for an infinitely periodic grating, it can still be very useful for designing finite length gratings. With this approach, one can quickly access how the performance of the grating is altered by various design parameters. For example, we can run a parameter sweep of the corrugation depth (see Optimization and Sweeps tab in Bragg_FDTD_unit_cell.fsp) and calculate the grating coupling coefficient

$$\kappa=\frac{\pi n_{g} \Delta \lambda}{\lambda_{0}^{2}}$$

where Δλ is the bandwidth, λ_{o} is the center wavelength, n_{g} is the group index at λ_{o}.

The results above require increasing the simulation time to 3750fs because the peaks in the spectrum are more difficult to distinguish for weaker gratings (with small grating depth).

### Coupling coefficient as a function of misalignment in the sidewall corrugations

In reference [1], the same FDTD method was applied to show how the grating strength can be tuned by varying the misalignment between the corrugations on the two sidewalls.

### Related publication

1. X. Wang, et al., "Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings", Opt. Lett. 39, 5519-5522 (2014).