In this example, the fundamental mode of a bent Silicon-on-Insulator (SOI) waveguide is analyzed with the FEEM solver. The modal fields, effective index, and losses are calculated as a function of bend radius and compared to simulation results from [1].

## Overview

Understand the simulation workflow and key results

The waveguide mode calculation is performed with the FEEM solver.

### Step 1: Straight Waveguide Mode Calculation

The effective index and mode profile of the fundamental mode of the straight waveguide are calculated by the FEEM solver.

### Step 2: Bent Waveguide Mode Calculation

The effective index and mode profile of the fundamental mode of a waveguide with a bend radius of 1.5 um are calculated. To help the FEEM solver find the bent waveguide mode, the effective index of the straight waveguide mode from step 1 is used as a starting point for the effective index search.

### Step 3: Bend Radius Parameter Sweep

A parameter sweep is used to obtain the real and imaginary parts of the effective index of the fundamental mode as a function of bend radius. Similar to step 2, the straight waveguide mode's effective index is used as a starting point for the FEEM solver's effective index search.

## Run and Results

Instructions for running the model and discussion of key results

### Step 1: Straight Waveguide Mode Calculation

- Open the simulation file feem_bent_waveguide.ldev .
- Open the script run_feem_simulation.lsf .
- Verify that the
`bent_flag`

variable is set to`false`

in line 5 of the script. - Run the script.

The script run_feem_simulation.lsf will run the simulation of a straight waveguide, plot the mode profiles, and print the effective index and loss results for mode 1 to the Script Prompt.

Viewing the mode field profiles allows us to determine the mode numbers of specific modes. If there are multiple modes found by the solver, visualizing the mode profiles can help determine the mode number of the mode of interest. The magnitude of the electric field of the profile of the first mode found in this example is shown below:

The individual components of the electric field profile can also be viewed in the ** Visualizer ** by changing the ** Vector operation ** to confirm the polarization of the mode.

The mode calculation returns an effective index of 2.3904. As this is a straight waveguide with no absorbing materials the mode solver results have no loss.

### Step 2: Bent Waveguide Mode Calculation

- Change the
`bent_flag`

variable to`true`

in the run_feem_simulation.lsf script. - Run the script.

When the ```
bent_flag
```

variable is set to ```
true
```

the run_feem_simulation.ldev script will modify the simulation file and run a simulation of a bent waveguide with a bend radius of 1.5 um. The electric field profile of the bent waveguide mode is shown below:

As expected, the mode field has moved to the outside of the waveguide bend compared to the straight waveguide mode. The calculated effective index is 2.39784 + 7.89557e-06i with a loss of 278.001 dB/cm.

### Step 3: Bend Radius Parameter Sweep

- With the simulation file feem_bent_waveguide.ldev open, run the script waveguide_bend_sweep.lsf .

This script will run the "bend radius sweep" parameter sweep and plot the real and imaginary effective index results as a function of waveguide bend radius. For comparison, results of simulations using a modified finite-difference (FD) method and a film mode matching (FMM) method from [1] are included in the plot as well:

The results from FEEM have good agreement with the results from [1] for the real part of the effective index and the imaginary index values for lower bend radius, but the FEEM results diverge from the published results for high bend radius. However, the imaginary effective index values at this bend radius are very low and are within the expected floating-point accuracy of the FEEM solver.

## Important Model Settings

Description of important objects and settings used in this model

### Boundary Conditions

PML boundary conditions are required for absorbing the radiating fields of the bent waveguide mode to calculate the bend loss. Typically, because the fields are radiating towards the outside of the bend. Metal (closed) boundary conditions should be used for this boundary because they require less simulation time than PMLs.

### Bend Radius and Bend Location

The ** bend radius ** is defined as the distance between the center of curvature and the **bend location**. The choice of ** bend location ** is technically arbitrary, however typically either the center or outside edge of the waveguide core is chosen. Either can be chosen for a given geometry, as long as the ** bend radius ** takes into account where the bend location is placed.

Note that, because the choice of ** bend location ** is arbitrary, the ** bend radius ** for a given geometry is also arbitrary. As the linear effective index depends on the ** bend radius ** , it is also arbitrary. The choice of ** bend location ** won’t change the physically relevant quantities, which are the mode field profiles and the angular effective index. For more information on this issue see Solving bent waveguides in FDE and FEEM .

### FEEM Region Span

If the PML boundaries of the FEEM simulation region are too close to the waveguide core it could introduce artificial gain/loss to the calculated waveguide modes. Plotting the electric field profile in log scale can help determine if the fields have decayed enough before reaching the boundaries. Ideally the electric field magnitude should be reduced down to around 10 ^{ -4 } times the maximum value at the boundaries. Ultimately, convergence testing should be used to determine if the FEEM region span is large enough.

### Index to Search Near

The ** n ** property in the FEEM solver's ** Modal Analysis ** tab sets the effective index value where the FEEM solver will begin its search for waveguide modes. For straight waveguides the maximum index in the simulation region is a good value to start with. However, for bent waveguides it is best to start with a value closer to the effective indices of the modes of interest to help the FEEM solver find them. The effective index of the mode of interest in a straight waveguide is often close enough, which is why it is used in this example.

For waveguide modes with a complex effective index (such as bent waveguide modes), it can also be useful to start the search for the effective index with an index with a small imaginary component, for example to a value of 2.3904 + 1e-5i.

### Number of Trial Modes

If the ** auto remove pml modes ** setting of the FEEM solver is set to ** true ** , the solver will automatically detect unphysical modes that overlap with the PML regions and remove them from the mode list. This is why the number of reported modes may be less than the ** number of trial modes ** setting of the FEEM solver. If only PML modes are found, no modes will be reported by the FEEM solver. In this case the ** number of trial modes ** setting should be increased until physical waveguide modes are found. Increasing the ** number of trial modes ** setting will increase the simulation time.

## Updating the Model With Your Parameters

Instructions for updating the model based on your device parameters

### Waveguide Geometry

The waveguide width and height can be changed by adjusting the geometry properties of the "waveguide" geometry object, for the example the ** x span ** and ** y span ** . Additional geometry objects can be added to make more complex waveguide geometries.

### Waveguide Materials

The materials of the waveguide, substrate, and cladding can be changed using the ** material ** property of the geometry objects. Additional materials can be added to the simulation from the Material Database.

### Wavelength

Unlike the FDE solver the FEEM solver has no ** Eigenmode Analysis ** window. The ** wavelength/frequency ** of the calculated modes is a property of the FEEM solver itself, found under the ** Modal Analysis ** tab of the ** Edit Finite Element Eigenmode Solver ** window.

### Bend Radius

The bend radius is set using the ** bend radius ** property of the FEEM solver. Note that the ** bend radius ** must be greater than the distance between the ** bend location ** and the boundary to the inside of the waveguide bend (the x min boundary in this example). In other words, the center of curvature cannot be inside the FEEM simulation region.

## Additional Resources

Additional documentation, examples and training material

### References

- Jinbiao Xiao and Xiaohan Sun, "Vector analysis of bending waveguides by using a modified finite-difference method in a local cylindrical coordinate system", Opt. Express 20, 21583-21597 (2012)