In this topic, we demonstrate how to simulate fiber Bragg grating (FBGs) using MODE' eigenmode expansion (EME) solver.

## Simulation setup

The FBG is constructed with an effective index of 1.5, and a periodic variation of 1e-3 in the refractive index of the core of a step-index fiber. The refractive index contrast, as well as the pitch and duty cycle of the grating, can be tailored so that a specific wavelength of light can be reflected while the rest of the spectrum is completely transmitted, making the FBG an efficient optical filter.

For periodic structures, only 1 unit cell of the geometry needs to be defined. In the [[fbg.lms]], the EME solver covers a single unit cell of the FBG as shown below.

As shown above, a port is set up at each end of the solver region to calculate the transmission and reflection into the fundamental mode. Under the EME setup tab, we define two cell groups for the EME solver, one covering the high index and the other covering the low index region. Since the refractive index and geometry are uniform within each cell group, we only need to use one cell for each cell group. We will use four modes for each cell group in the EME calculation.

To set the periodicity of the FBG, we will define one periodic group under the "periodic group definition" table. The start and end cell groups are set to '1' and '2' respectively, and the number of periods is set to '20000.' This means that the unit cell (composed of 2 cell groups) will be propagated 20000 times, and the final length of the FBG will be 1cm.

Since EME is a frequency-domain method, we will need to run one simulation for each wavelength of interest. To simulate the full transmission/reflection spectrum, a parameter sweep is used to scan the wavelength from 1.495um to 1.504um. The user s-matrix result from the EME region is stored as the result.

## Results

Once the parameter sweep is completed, the S parameter result can be plotted in the Visualizer by right-clicking on the parameter sweep object and Visualize->S. Since there are two ports (with one mode to track at each port), S11 and S21 will be the reflection and transmission of the FBG, respectively. The figures below show the S21 element of the user s-matrix as a function of the wavelength. To increase the resolution, simply increase the "number of points" in the parameter sweep.

Fiber Bragg grating transmission calculated using (Left) parameter sweep with 100 points and (right) wavelength sweep feature with 5000 points.

One can also obtain similar results using 100 points with the wavelength sweep feature in the EME Analysis window. This technique is much faster compared to parameter sweep object, and a wavelength sweep with 5000 points takes only few seconds. To do this, set the wavelength span in the wavelength sweep section and then press wavelength sweep. Once the calculation is finished, use "visualize wavelength sweep" button and then select Abs^2 in the visualizer as is explained above.The EME method is ideal for FBGs because increasing the number of periods or length of the FBG device does not increase the simulation time. A 1cm FBG will be very difficult to simulate with FDTD-based methods due to the amount of computational time and memory required.

### See also

Bragg Grating Initial Design with FDTD