In this example, the mode losses due to radiation into a high index substrate for the fundamental mode of an SOI waveguide will be calculated with the FEEM solver. This requires the use of PML boundary conditions, which were added to the FEEM solver in release 2021 R1.4. The results with different mesh settings will be compared to simulation results from the literature through convergence testing.

## Overview

Understand the simulation workflow and key results

The loss calculation is performed with the FEEM solver.

### Step 1: Initial Mode Calculation

Typical values are used for simulation parameters, including
**
edges per wavelength
**
and
**
polynomial order
**
, for an initial estimate of the effective index.

### Step 2: Convergence Test

Convergence testing
is performed using a parameter sweep of the mesh parameters
**
edges per wavelength
**
and
**
polynomial order
**
to estimate the relative accuracy of the results.

## Run and Results

Instructions for running the model and discussion of key results

### Step 1: Initial Mode Calculation

- Open the simulation file [[waveguide_substrate_loss.ldev]].
- Run the script [[waveguide_mode_properties.lsf]].

The script [[waveguide_mode_properties.lsf]] will set up the simulation properties, run the simulation, 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. In this case, because the
**
number of trial modes
**
setting of the FEEM solver is set to 1, only a single mode (the fundamental TE mode) is found. In general, 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 mode profile found in this example is automatically plotted by the script, as shown below:

In this initial simulation, the typical starting values of
**
edges per wavelength
**
= 1 and
**
polynomial order
**
= 3 are used (see the
Important Model Settings
section for more information on these parameters). Due to the symmetry of the waveguide and the TE mode, we can reduce the size of the simulation region by placing the x min boundary at the center of the waveguide and using PEC conditions at this boundary.

The mode calculation with these simulation settings results in an effective index of 2.40759 + 2.7542e-8i, with a loss of 0.969744 dB/m.

### Step 2: Convergence Test

- Run the script [[substrate_losses_convergence_test.lsf]] in FEEM with the simulation file [[waveguide_substrate_losses.ldev]] open.

In this example a parameter sweep for the convergence has been created, named “convergence test sweep”. The “convergence test sweep” is a nested sweep of the
**
edges per wavelength
**
property (with 1, 3, and 5 edges per wavelength) and the
**
polynomial order
**
property (from 3 to 10). The sweep is already populated with the results in the example file, but you can optionally run the parameter sweep by setting the
```
run_sweep
```

variable to
```
true
```

in line 2 before running the script.

The [[substrate_losses_convergence_test.lsf]] script will use the parameter sweep results to calculate the relative error \(\sigma_i\) at each step \(i\) as defined by

$$\sigma_i = \left| \frac{x_i - x_{ref}}{x_{ref}} \right|$$

Where \(x_i\) is the calculated real or imaginary part of the effective index at each step and \(x_{ref}\) is the reference effective index value of 2.4123720 + 2.91348e-8i [1]. The script will also plot the results on a log scale:

From these results we can see that the increasing either the polynomial order or the edges per wavelength can increase the accuracy of the simulation, as expected. At the highest level of accuracy in the sweep, with
**
edges per wavelength
**
= 5 and
**
polynomial order
**
= 10, the imaginary effective index result is within 0.005% of the reference value.

## Important Model Settings

Description of important objects and settings used in this model

### Edges Per Wavelength

The FEEM solver works on an unstructured triangular mesh. A mesh with smaller triangles generally leads to a more accurate representation of the geometry and the fields and therefore gives more accurate results. However, more elements also increase the simulation time. Since the cross section of a mode typically varies slowly away from material interfaces, a small value like 1 or even less is good starting point.

### Polynomial Order

On each triangle, the solver expands the electric field into polynomials up to a user-specified order. For smooth fields, a higher order leads to a more accurate solution but it also increases the computational cost. A
**
polynomial order
**
of 3 is typically a good initial value.

### Sigma

The Perfectly Matched Layers introduce absorption in a finite shell around the simulation region. The
**
sigma
**
parameter defines the strength of the absorption. If the value is too small, radiation can still reach the outer, reflective boundary of the simulation domain. If
**
sigma
**
is very large, it can lead to numerical reflections at the inner interface of the shell and pollute the solution. The default value of 5 usually is a good compromise but depending on the mode, the distance of the structure and the thickness of the shell, changing the value can lead to better results.

### Index to Search Near

When a high index substrate is included in the simulation region many of the modes will be substrate modes, especially if the
**
index to search near
**
is set to
**
max index
**
. To ensure that the waveguide modes are found, it is best to set the
**
index to search near
**
to the approximate value of the waveguide mode effective index.

For waveguide modes with a complex effective index (in other words, modes with loss), it can also be useful to start the search for the effective index with an index with a small imaginary component. This can be set using the
**
n
**
property in the FEEM solver’s Modal Analysis tab, for example to a value of 2.41 + 1e-5i.

### Auto Remove PML 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 that 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.

## Updating the Model With Your Parameters

Instructions for updating the model based on your device parameters

### Waveguide Geometry

The waveguide width and height can 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. Note that with a new geometry different simulation settings may be required, so convergence testing will need to be performed again. It is also important to ensure that the substrate/cladding are extended through the PML regions, so the material properties are correct in those regions.

### 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

The wavelength/frequency is set in the "FEEM" property window, under the
**
Modal Analysis
**
tab.

## Taking the Model Further

Information and tips for users that want to further customize the model

### Convergence Testing With No Reference Value

In most cases, there won’t be a known reference value with which to compare your simulation results. In this case there are other methods for estimating the error for convergence testing, as described on the page Convergence testing process for FDTD simulations .

## Additional Resources

Additional documentation, examples and training material

### Related Publications

- Modelling leaky photonic wires: a mode solver comparison, P. Bienstman et al., Optical and Quantum Electronics 38, 731–759 (2006)

### See Also

- Waveguide (FEEM)
- Thermally tuned waveguide (FEEM)
- Lithium Niobate Nonlinear Thermal Waveguide (FEEM)
- Convergence testing process for FDTD simulations