Calculate the response of a 1x8 arrayed waveguide grating (AWG) working as a demultiplexer. An INTERCONNECT compact model is initially used for quick analysis. Component-level simulations using varFDTD are carried out for more realistic results. The final design can be exported to a GDS file for fabrication.

## Overview

Understand the simulation workflow and key results

Design of AWG typically requires combinations of simulation tools as well as analytical models. We start with the eigenmode solver to calculate the modal properties of a single waveguide and a slab. This is followed by the varFDTD simulation to further characterize the properties of beam that gets diffracted when it enters the slab from a waveguide. The parameters extracted from these simulations as well as those obtained analytically are used in the ensuing INTERCONNECT simulations based on a simplified compact model, and a more realistic component-level simulations in varFDTD.

### Step 1: Effective and group indices of waveguide and slab (FDE)

We calculate the effective and group indices of the waveguide and slab using the eigenmode solver (FDE). These results will be used as input parameters in the next steps for INTERCONNECT and varFDTD simulations.

### Step 2: Field amplitude versus angle (varFDTD)

To study the coupler regions, we create a simplified model in varFDTD to study the diffraction properties of the waveguide modes in the slab region. This will provide the electric field amplitude distribution as a function of angle.

### Step 3: AWG compact model (INTERCONNECT)

We make a parameterized INTERCONNECT compact model for AWG. This INTERCONNECT compact model provides a way to quickly simulate a variety of different AWGs. The model is parametrized where, for example, the number of output channels or array waveguides can be modified.

### Step 4: AWG simulations in blocks (varFDTD) and GDSII export

The AWG can be simulated in blocks using varFDTD. The couplers can be simulated by varFDTD and the arrayed waveguides can best be handled analytically. The advantage of this approach is that it only requires two varFDTD simulations of the coupler regions. The results are compared with step 3, and geometry is exported into a GDS file.

## Run and results

Instructions for running the model and discussion of key results

### Step1: Effective and group indices

- Open the Arrayed_waveguide_grating_slab_study.lms. Make sure that FDE is activated.
- Make sure that “calculate group index” is enabled from the Mode Advanced Options window. Set the wavelength to 1.55 μm and calculate modes.
- Repeat the same with Arrayed_waveguide_grating _waveguide_study.lms

From the first simulation, an effective index (neff) of 2.83 and a group index (ng) of 3.74 are obtained for the fundamental mode of the slab. The second simulation gives neff = 2.39 and ng = 4.2 for the fundamental mode of the waveguide. These index values will be used in the INTERCONNECT (Step 3) and varFDTD (Step 4) simulations. The above simulation files also include some disabled objects which will be used for the characterization of the grid dispersion properties of the waveguide and slab using varFDTD (see Appendix.)

### Step2: Field amplitude versus angle

- Open the Arrayed_waveguide_grating _coupler_study.lms and run it.
- Load the script file Arrayed_waveguide_grating _coupler_study.lsf and run it.

This file will extract the magnitude of the E field as a function of angle. It will use this to reconstruct the E field, in amplitude and phase, in the near field. It will then compare this to the simulated result using the farfieldexact projection. The plot below shows comparison of |E| and angle at a distance of 50 µm obtained from the two approaches.

The field amplitude at arbitrary positions can be calculated using the farfield2d script command and scaling them accordingly. This approach is more efficient than using the farfieldexact since you need to calculate the farfield only once. Further information about farfield scaling can be found here.

The variation of |E| vs theta at different wavelengths is shown below, from which we realize that it is reasonable to represent the value of |E| at a single wavelength of 1550 nm.

The results will be printed in the script prompt and are copied into the Simulation-Setup tab of AWG compact model in INTERCONNECT.

### Step3: INTERCONNECT compact model

- Open the Arrayed_waveguide-grating_compact_model.icp.
- Right click on the AWG element on the schematic editor and select Edit. In the property editor window modify the effective and group indices of the waveguide and slab based on the results from step 1. The |E| as a function of theta in each waveguide is also imported into Simulation-Setup tab from step 2. Run the simulation file.
- Use the script below to plot the output of AWG for all 8 ports.

clear;

num_output_port = 8;

leg = cell(num_output_port);

for (i=1:num_output_port){

eval("

temp= getresult('ONA_1','input "+num2str(i)+"/mode 1/gain');

gain"+num2str(i)+"=temp.getattribute('TE gain (dB)');

");

plot(temp.wavelength*1e9,temp.getattribute('TE gain (dB)'));

leg{i}="port "+num2str(i);

holdon;

}

holdoff;

setplot("x label","wavelength (nm)");

setplot("y label","Re ( TE gain(dB) )");

legend(leg);

The model consists of a total of 32 output ports, allowing the number of channels (Nchannel) to be set from 1 to 32. It is based on the analytical model in the Smit et al. [ref. 2]. In the input coupler, the E field amplitude at the entrance of each arrayed waveguide is calculated using the |E| vs theta data from step 2. Coupling is estimated by the waveguide width and the waveguide_acceptance_fraction. The light is propagated through the array waveguides using the values of neff and ng from step 1 and estimated loss. At the output coupler, the coherent sum of the contributions from all the Narray waveguides to the output waveguides is calculated, again using the previous |E| vs theta data. The phase is calculated based on the propagation distance. The results are smoothed based on the finite waveguide input width.

The following plot shows the result for 8 channels and is in good agreement with figure 3 of Li et. al. [1]:

### Step4: varFDTD simulations in blocks

** Input coupler: **

- Open the Arrayed_waveguide_grating_varFDTD.lms file that contains the basic AWG setup. Edit the AWG analysis group, set simulation_type to 1 (input coupler) and the taper_waveguides setting to 0. Run the simulation.
- Once the simulation is finished, run the AWG analysis group and plot the results. (The calculated results are saved to the file varFDTD_input_coupler_results.mat and will be used to setup and analyze the second stage.)

The result T shows the transmission in each of the Narray (array number) waveguides vs lambda ( left). As expected, we have the maximum transmission at the central waveguides, however, the function is not smooth. Some care could be taken to optimize the input coupler design. The T_total results gives the total transmission by summing over all Narray waveguides as a function of wavelength (middle). The result amplitude is calculated by averaging the amplitude across all wavelengths (right).

The phase and group delay vary as a function of Narray. Variations in these quantities can be mainly attributed to the grid dispersion. See the “grid dispersion in FDTD and varFDTD” in the appendix section below for more details.

** Output coupler: **

- In the same simulation file, set “simulation_type” to 2 and “taper waveguide to 1”. Run the simulations.
- Once the simulation is finished, run the AWG analysis group and plot the results that shows transmission in each output waveguide (8 channel here).

This simulation will use 24 different waveguide sources, one for each arrayed waveguide. The source pulses used in each waveguide will be offset precisely by the desired group delay difference in each waveguide, which is

$$ \Delta t = \Delta L \frac{n_g}{c} $$

where \(\Delta L\) is the length difference of the arrayed waveguides. This results in the target group delay difference in each waveguide. We can optionally include the extra group delay difference as well as the phase calculated in the “input coupler” section of step 4. However, since those were mainly a result of FDTD grid dispersion, we will not include them here. The analysis will also include the effect of the transmission vs wavelength of the first stage.

The simulation loads the input_results.mat data saved from “simulation type=1” to set the source amplitude and phase accordingly.

** Whole device: **

- Set the simulation type to 3 to draw the whole device including the arrayed waveguide section and get it ready for export to a GDS file. This simulation_type setting can be also used for brute force simulation of the entire AWG.

Note that running a simulation including the entire AWG can be very slow and suffer from large cross talk due to errors in group delay and phase, which again can be attributed to the grid dispersion.

Please see this page to learn more about how to export the geometry into a GDSII file.

## Important model settings

Description of important objects and settings used in this model

INTERCONNECT compact model: The design parameters such as delta_L and output waveguide spacing are calculated from the input parameters using the design principles in Smit et al [ref. 2]. These design parameters are listed under the “Property Editor” tab of the “Edit Scripted_1” window (right-clicking the “Scripted_1” object in the Schematic Editor and select “Edit”) and the script defining the behavior of the device can be found under the “Simulation” tab.

VarFDTD vs FDTD: The varFDTD method can correctly simulate the group index of slabs and waveguides. Moreover, near to far field projections use the appropriate Hankel functions to handle propagation of light in a slab, making it easy to study the diffractive properties of waveguide modes without having large simulation regions.

The coupler regions of AWGs are, for the most part, slabs of silicon and therefore the varFDTD method is an excellent choice to simulate these regions. Final verification by full 3D FDTD may be desired.

Tapered ending of waveguides: The input and output waveguides were intentionally tapered and set to terminate before the PML. This was necessary to prevent the simulation from running much longer and becoming more prone to numerical instability as a result of the dispersive material intersecting the PML. The “AWG” analysis group includes a parameter called “taper_waveguide” in case you want to test the untapered waveguides.

## Updating the model with your parameters

Instructions for updating the model based on your device parameters

Analytical model: We use an analytical model based on the Smit et. al. design [ref 2] to simulate AWG in INTERCONNECT. If you want to update the Smit model with your own, you might need to update the script as well as the associated parameters. The most important change would be related to how the input and output couplers are handled. In most cases, this could be modified simply by changing the data that defines |E| vs theta. If the new analytical model requires some update or additional parameter values to be passed on from the initial steps 1 and 2, then you might need to update the simulation settings for those steps as well.

“AWG” analysis group: The AWG in varFDTD is parameterized by the analysis group “AWG”, which defines the geometric structures, the mode sources, the frequency domain monitors and the mode expansion monitors. The design follows the basic principles of Smit et al. and the setup script can be modified to accommodate other designs for the couplers. For example, if you want to incorporate tapers to the input and output waveguide sections of the couplers, you need to modify the setup parameters as well as the script associated with them. Note that the script is complex because the varFDTD region, the monitors, sources, etc, need to be aligned with all the waveguides. If you want to compare the results from step 3 (INTERCONNECT) and step 4 (varFDTD), make sure the values of the parameters used in each simulation step remain the same.

## Taking the model further

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

Field amplitude at each waveguide: In step 2 we calculate the field distribution as a function of angle. One can directly calculate the field amplitude in each arrayed waveguide by using the first part of step 4 and then import the results into INTERCONNECT model.

Updating the geometry: For complex waveguide geometries, such as tapered waveguides in the coupler-arrayed waveguides region, the results of INTERCONNECT compact model may not be accurate as it assumes a simple diffraction in the output coupler region. For cases where the etch depth varies in the coupler region, an FDTD simulation might be necessary to correctly calculate the field distribution in each arrayed waveguide.

S-parameter of AWG: We can also extract S parameters for the entire AWG after running the input and output couplers from step 4 to construct an S parameter model in INTERCONNECT. After exporting, these can be loaded into an N port S parameter element. This approach is useful if you have your own custom AWG design that you want to use in a circuit, and compact model cannot predict the results.

Optimizing the geometry: There are many different design parameters to optimize for improved performance of the AWG. This will require modifying the AWG analysis group script to reflect the new geometry. Coupler geometry could be optimized to minimize Fabry-Perot effects. Using tapered waveguide in the arrayed waveguides can be also considered for further improvement in the coupling between the couplers and the arrayed waveguides.

## Additional resources

Additional documentation, examples and training material

### Related publications

- Hongqiang Li et al., “Investigation of Ultrasmall 1×N AWG for SOI-Based AWG Demodulation Integration Microsystem”, IEEE Photonics Journal, 7, 2015.
- Meint K. Smit et al., “PHASAR-Based WDM-Devices: Principles, Design and Applications,” IEEE Journal of Selected Topics in Quantum Electronics, 2, 1996.

### See also

### Related Lumerical University courses

## Appendix

Additional background information and theory

### Grid dispersion in FDTD and varFDTD

The speed of light on the FDTD mesh is not correct for finite dx and dt, and the speed of light becomes anisotropic. This causes the grid dispersion in varFDTD and FDTD. The documentation of the script function getnumericalpermittivity explains more details on this effect.

AWGs are devices that rely on precise relative phase and group delays between different optical paths through the device. The challenge is that in an AWG light propagates at many different angles, and therefore the precise phase of the light (and the group delay) will be incorrect. Of course, this converges as dx, dy, dz and dt are reduced, but any error in an AWG can be significant because the light propagates many hundreds of wavelengths.

### Grid dispersion and slab modes:

First, let’s consider the effect of grid dispersion in varFDTD for the slab modes. The file Arrayed_waveguide_grating_slab_study.lms will allow us to measure the slab effective index and group index as a function of angle.

- Select the FDE solver and calculate the modes at 1.5 microns.
- Select the Frequency analysis tab and enable the “track selected mode”. Then perform a frequency sweep from 1.5 to 1.6 microns and plot the group index.

We see that the effective index at 1.55 microns is about 2.86 and the group index is about 3.74. Note that for the FDE solver, we chose to fit sampled materials with the multi-coefficient model (MCM) to avoid discontinuities when calculating the group index:

- Activate the varFDTD solver. Edit the varFDTD Solver and set the bandwidth in the Effective Index tab to “narrowband”. In other words, we have a non-dispersive effective index model where the effective index matches the slab index at 1.55 microns.
- Run the varFDTD solver and plot the simulated slab effective (neff vs theta) and group indices (ng vs theta) as a function of angle of propagation with respect to the cartesian x axis using the “slab_study” analysis group.

As is shown below both the slab effective index and group index are dependent on angle, but the group index is completely incorrect.

- Switch back to layout mode and reset the bandwidth of the varFDTD solver in the Effective Index tab to broadband. In the materials explorer, we see that we now have a dispersive effective material. In addition, we see that it has a small, non-zero imaginary part. This is necessary to create dispersion.
- Run the varFDTD solver and plot the simulated slab effective (neff vs theta) and group indices (ng vs theta) as a function of angle of propagation with respect to the cartesian x axis using the “slab_study” analysis group.

As is shown below, while neff and ng still have angular dependence, the slab group index is now correct.

### Grid dispersion and waveguide modes

We can now perform a similar analysis of the effective and group indices of waveguides with varFDTD.

- Open the file Arrayed_waveguide_grating_waveguide_study.lms and make sure the FDE solver is active. Also, ensure that the “rotation angle” property of the waveguide_study analysis group is 0.
- Run the FDE simulation and calculate the modes at 1.5 microns and then perform a frequency sweep from 1.5 to 1.6 microns. We see the following plots for effective index and group index. At 1.55 microns, the effective and group indices are approximately 2.4 and 4.2 respectively.

Expand the waveguide_study analysis group and set the varFDTD solver to active. Also ensure that the bandwidth property in the Effective index tab of the varFDTD solver is narrowband.

- Run the sweep object called “sweep” and plot the results for neff and ng.

This will give the result of a traditional 2D FDTD effective index method, and we can see that the waveguide group index is completely incorrect.

- Expand the waveguide_study analysis group and set the varFDTD bandwidth to broadband.
- Run the sweep object called “sweep” and plot the results for neff and ng.

You can see that the varFDTD simulation results are very close to the correct value calculated by the FDE solver at 1.55 microns. You can see some variation of these results with angle of the waveguide as well. This variation is not as smooth as the slab neff and ng which are related only to FDTD grid dispersion. In this case, there is the added effect of the waveguide meshing leading to some staircasing effects that are not eliminated by the conformal mesh technology.

Even though varFDTD with the conformal mesh technology can reproduce accurately the effective index and group index, the small angular dependence of these quantities will lead to challenges when simulating the AWG. For example, an AWG with 24 arrayed waveguides and a differential length of 30 microns has a length difference between the first and last waveguide of 720 microns. An error of only 0.01% in effective index between these different waveguide paths, can result in a phase error of about 40 degrees. These types of errors are enough to ruin the operation of the AWG by yielding larger than expected cross talk.