This example demonstrates the design and simulation of a coupled-resonator optical waveguide (CROW) filter using the FDE, FDTD, and INTERCONNECT solvers. Starting with specific design objectives, simulations will be used to explore the design parameter space and determine the physical parameters required to obtain those design objectives.

## Overview

Understand the simulation workflow and key results

This example will focus on the design of a CROW filter with two identical rings, each with a coupling coefficient \(\kappa_1\) between the rings and the bus waveguides, and coupling coefficient \(\kappa_2\) between the rings themselves, as shown in the image below.

The CROW filter model in this example will also include thermal tuning, which can be used to adjust the filtered wavelengths. The goal of the design is to create a filter with an FSR of 79.5 GHz and a center wavelength at 1550 nm. The waveguide cross-section geometry is assumed to be constrained by the fabrication process, so the physical parameters that must be determined include the ring radii, the gap distance between the rings, and the gap distance between the rings and the bus waveguides. The thermal tuning voltage required to obtain a filtered wavelength at 1550 nm will be determined as well.

### Step 1: Waveguide FDE simulation

First, the properties of the fundamental TE mode, including effective index \(n_{eff}\) and group index \(n_g\), are determined using the FDE solver. The group index is used to determine the ring radius \(r\) required to obtain the desired FSR using the formula

$$FSR=\frac{c}{n_g 2π r}$$

### Step 2: INTERCONNECT coupling coefficient simulation

While it would be possible to perform FDTD simulations of the entire CROW structure, time-domain simulations of resonator structures require very long simulation times and are not practical for performing parameter sweeps. For this reason, a parameterized compact model of the CROW filter will be created using INTERCONNECT elements and used to characterize the dependence of the device characteristics on various design parameters, such as the effective and group indices, ring radius, and coupling coefficients.

This INTERCONNECT model will first be used to determine the coupling coefficients \(\kappa_1\) and \(\kappa_2\) required to obtain the desired performance.

### Step 3: INTERCONNECT thermal tuning simulation

Next, a parameter sweep of the voltage applied to the thermal tuners will be used to determine the voltage required to obtain a resonant wavelength of 1550 nm, assuming a tuning efficiency of

0.1 nm/V\(^2\).

### Step 4: Coupler region FDTD simulation

Lastly, FDTD simulations of the coupling regions will be used to determine the gap distances required to obtain the desired coupling coefficients \(\kappa_1\) and \(\kappa_2\).

## Run and Result

Instructions for running the model and discussion of key results

### Step 1: Waveguide FDE simulation

- Open the file [[CROW_waveguide.lms]] in MODE Solutions.
- Run the simulation.

The results of the simulation show that \(n_{eff}\) = 2.566 and \(n_g\) = 3.893. Using the FSR formula mentioned in the Overview section, the required ring radius is determined to be \(r\) = 154 μm. As this is a large radius, we will assume the bent waveguide mode characteristics are similar to those of the straight waveguide. This can be verified using the bent waveguide solver (see the “Important model settings” section for more information).

### Step 2: INTERCONNECT coupling coefficient simulation

- Open the file [[CROW_photonic_compact_model.icp]] in INTERCONNECT. The CROW filter compact model parameters can be adjusted in the models “Property View”. The effective index, group index, and ring radius from the previous step have already been set as the parameters of this model. The individual elements that make up the CROW filter compact model can be viewed by right-clicking on the CROW filter element and selecting “Expand”. The script used to parameterize the model can be viewed by right-clicking on the CROW filter element and selecting “Edit”, then selecting the “Script” tab in the edit dialog window.
- Run the parameter sweep "kappa2" to sweep over both coupling coefficients.
- The results of the sweep can be viewed in a Visualizer window. With the completed sweep selected, select both the “gain_through” and “gain_drop” results in the Result view, right-click and select “Visualize > New Visualizer” to open both results in a Visualizer window. Under “Plot types” select “line”, and under “Parameters” set both “kappa1_squared” and “kappa2_squared” to “Slice” and change the x-axis from “frequency” to “wavelength”. Zoom in on the peak near 1550 nm. Under “Plot settings” turn on “Hold Camera”. The dependence of the transmission spectrum on the coupling coefficients can be explored using the sliders under the “Parameters” section.

Based on the gain spectra, we will select \(\kappa_1^2\) = 0.13 and \(\kappa_2^2\) = 0.0047 as the nominal values for our design. Here is the gain spectrum with those values:

It is worth noting the speed of the INTERCONNECT simulations during this sweep. Performing the same sweep over two variables in an FDTD simulation would take many hours, while INTERCONNECT can perform the sweep in minutes.

### Step 3: INTERCONNECT thermal tuning simulation

- Set the "kappa1_squared" and "kappa2_squared" parameters of the CROW filter compound element to the nominal values determined in the previous step.
- Run the "thermal_tuning" parameter sweep to obtain the gain spectra as a function of the voltage applied to the thermal tuners.
- Visualize the sweep results.

From the results of this sweep we can see that applying a voltage of approximately 0.95 V to the thermal tuners shifts the resonance to 1550 nm:

Optionally, the coupling coefficient sweeps can be repeated to verify that applying this voltage does not change the optimum coupling coefficients for this design.

### Step 4: Coupler region FDTD simulation

- Open the file [[directional_coupler.fsp]]. This file contains a parameter sweep used to determine the coupling coefficients as a function of gap distance, for both the ring-bus and ring-ring coupling regions. Due to the length of time required to perform FDTD simulations, the sweep results are already included in this file.
- Run the script [[calculate_gaps.lsf]]. This script calculates the distance required to obtain the desired coupling coefficients based on the sweep results and prints them to the Script Prompt. The script also creates a plot of the sweep results:

The output of the script shows that the gap distances required for our design are approximately 410 nm for the ring-bus coupler and 630 nm for the ring-ring coupler.

## Important Model Settings

Description of important objects and settings used in this model

### Step 1: Waveguide FDE simulation

**
Mesh
**
: In general, a finer mesh produces more accurate results for the FDE solver. Convergence testing can be used to determine if the mesh is fine enough.

**
Simulation span
**
: When calculating modes with the FDE solver, the simulation span must be large enough for the modal field profiles to sufficiently decay before reaching the boundaries. As with the mesh, convergence testing can be used to determine if the simulation span is large enough.

**
Boundary conditions
**
: The boundary conditions of the FDE solver can be set to metal for a well-confined mode, however, PML boundaries must be used to calculate radiation or bend losses.

**
Waveguide bend radius
**
: The properties of a waveguide mode, including the effective index, group index, and losses will depend on the bend radius of the waveguide. Bent waveguides can be simulated using the FDE solver. Once the ring radius has been determined, the waveguide properties could be recalculated to verify that they do not change significantly in the bent waveguide of the ring compared to a straight waveguide.

### Step 2: INTERCONNECT coupling coefficient simulation

**
Loss
**
: The properties of the filter will depend on the waveguide loss. While bend losses can be calculated using the FDE solver, it can be difficult to determine loss from other sources, for example, scattering loss, in a simulation. The waveguide losses are assumed to be 1 dB/cm in this example.

**
Dispersion
**
: In this example, it is assumed that the dispersion of the waveguide mode is zero, but it could be calculated by the FDE solver using a frequency sweep and included in the INTERCONNECT waveguide elements in the CROW filter compact model.

**
Sample rate
**
: If performing time-domain simulations in INTERCONNECT it is crucial that the sample rate is high enough, in particular for simulations involving resonators. For this CROW filter compact model, the minimum sample rate is \(2c/(\pi r n_g)\), where \(r\) is the ring radius and \(n_g\) is the group index.

### Step 3: INTERCONNECT thermal tuning simulation

**
Tuning efficiency
**
: The sensitivity of the waveguide mode effective index to the voltage applied to the heating elements is given by the tuning efficiency. In this example, the tuning efficiency is assumed to be 0.1 nm/V\(^2\). A more precise value could be calculated with the use of the HEAT thermal solver and the FDE or FEEM waveguide mode solvers. See our “Thermal switch” example for a demonstration of this type of calculation.

### Step 4: Coupler region FDTD simulation

**
Mesh in gap region
**
: When simulating waveguide couplers in FDTD, the gap between the waveguides must be properly resolved by the mesh. A mesh override region can be placed over the gap to ensure that there are enough mesh cells in the gap and that the mesh points align with the edges of the waveguides. A high accuracy mesh and mesh override region can automatically be applied by setting the “high accuracy mesh” model setup script variable to 1.

**
Extend structures through PML
**
: By default, the “extend structures through PML” setting is turned on, so structures adjacent to the PML boundaries are extended through the PML in the direction normal to the PML. However, if a waveguide intersects the PML at an angle this will cause reflections, so this setting is turned off for this simulation.

**
Number of PML layers
**
: To ensure that the light is completely absorbed by the PML boundaries it may be necessary to increase the number of PML layers. As this will increase the computation time required for the simulation, it is best to do this as part of your convergence testing.

**
Multifrequency mode injection
**
: Better broadband FDTD results could be achieved by using the multifrequency mode injection for the ports. However, because this example is mainly concerned with results near a wavelength of 1550 nm, this option was not used in this simulation. This option can be enabled by setting the “multifrequency injection” model setup script variable to 1.

## Updating the model with your parameters

Instructions for updating the model based on your device parameters

When updating this example with your parameters, it is important to remember that there are multiple simulation files involved in this example. If you change the waveguide width, for example, this parameter must be changed in both the FDE and FDTD simulation files. The new effective index for this waveguide geometry must also be added to the INTERCONNECT simulation file. This will also change the required ring radius, which must be updated in the FDTD simulation file.

To change the parameters in the INTERCONNECT simulation file, update the properties of the CROW filter compound element. The compound element setup script will then automatically update the settings of the sub-elements inside the compound element. To modify the bandwidth or wavelength of interest, it may be necessary to modify the ONA element as well.

A model setup script is used in the FDTD simulation file, so changes to the parameters can be made in this script to automatically update the geometry and simulation objects. It may also be necessary to update the calculate_gaps.lsf script file if the wavelength or target coupling coefficients are changed.

No script is used to set up the FDE waveguide mode simulation, so the simulation object properties should be modified directly.

Some parameters that may need to be updated include:

### Wavelength

To update the wavelength of interest, the “Wavelength” in the Eigensolver Analysis window of the FDE simulation, the “lambda0” setting of the CROW filter compact model, and the “lambda0” variable in the calculate_gaps.lsf script must all be modified. The simulation bandwidth of the FDTD simulation must also be changed to include the wavelength of interest if it is outside the default bandwidth of 1.5 to 1.6 microns.

If the wavelength of an FDTD simulation is changed, it is also generally a good idea to double-check the material fits in the Material Explorer.

### Waveguide properties

To change the waveguide geometry or material, the geometry objects in the FDE simulation file should be updated directly. The geometry of the waveguide in the FDTD simulation is parameterized in the model setup script, so the setup script variables can be modified to change the waveguide geometry. However, for more complex waveguide geometries the setup script itself may need to be changed, for example, to add new geometry objects into the simulation.

### Tuning efficiency

The tuning efficiency of the thermal tuners can be calculated using HEAT and FDE/FEEM simulations and applied to the CROW filter compact model.

## Taking the model further

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

### Filter properties

The procedure described in this example can be followed to obtain different desired filter characteristics. Adjustments to the parameterized INTERCONNECT model can be made to easily explore the transmission spectra of various CROW filter configurations.

### Higher-order filters

Higher-order CROW filters can be simulated by extending the INTERCONNECT compact model from this example with more coupler and waveguide elements in different configurations.

## Additional Resources

Additional documentation, examples and training material