Part 1 of the ring resonator tutorial uses MODE to design and simulate a ring resonator. Free spectral range (FSR) and quality factor (Q factor) are key performance metrics for this silicon on insulator (SOI) based waveguide design targeting onchip communication applications. In Part 2, we will consider how to carry out the parameter extraction and Monte Carlo analysis process for this design. Part 3 does the final simulation and parameter extraction using a 3D FDTD simulation.
Part 1: Design and initial simulation using MODE
Part 2: Parameter extraction and Monte Carlo using MODE
Part 3: Final parameter extraction using FDTD
Learning objectives
In this example, the user will learn to:
 Use Mode Expansion Monitors to extract the parameters for interfacing with circuit level simulations in INTERCONNECT.
 Compare the S parameter results with 3D FDTD.
 Use the Monte Carlo Analysis feature to track the effect of fabrication errors on the free spectral range (FSR) of the ring resonator.
 This page contains 3 independent sections. The first section (Parameter extraction) describes how to setup the mode expansion monitors for parameter extraction. If you prefer to skip this section,the completed simulation files are provided on the first page of the tutorial. The second section describes how to use the S parameter results from the first section in a circuit level simulation in INTERCONNECT. The final section shows how to track the effect of fabrication errors on the free spectral range (FSR) of the ring resonator by performing Monte Carlo analysis.
Parameter extraction
Mode expansion monitors
We will start with the file ring_resonator.lms from part 1.
 Open the ring_resonator.lms file.
 Before adding the mode expansion monitor, please read the following page on the calculations behind mode expansion monitors: Mode Expansion Monitors.
 Add a mode expansion monitor by pressing on the arrow on the Monitors button and select the Mode expansion monitor from the pulldown menu. Set the properties according to the following table. (Note that you can add mode expansion monitors in layout or analysis mode, so it is not necessary to switchtolayout if the simulation has already been ran.)
tab 
property 
value 


name 
expansion 
Geometry 
monitor type 
Linear Y 

x (μm) 
4.2 

y (μm) 
3.6 

y span (μm) 
3 
z (um) 
0 

z span (um) 
2 
We have positioned this monitor directly in front of the MODE source, and we will use the fundamental mode of the top waveguide to expand the field at the 4 ports of the ring resonator.
 In the Mode expansion tab, select the fundamental mode for "Mode calculation". You can use the Visualize Mode Data button to study the field profile for this mode.
 In the "Monitors for expansion table", select the 4 power monitors we have set up at the 4 ports of the Ring Resonator as follows:
Plot results
Once the mode expansion monitor has been defined, you will see the list of results in the Result VIew panel. Multiselect the modal expansion results and select "Calculate". Once the calculations are complete, one can plot the results in the Visualizer.
Note that when the Visualizer first opens up, you will see a list of all the attributes of all the results. One can use the "Remove" button on the right side of the attribute panel to remove any unwanted attributes, keeping only the relevant ones. For a complete description of all the results from the mode expansion monitors, please refer to Mode Expansion Monitors.
S parameter calculations
In ring_resonator2.lms, the model analysis group in the provided premade simulation file has been set up to calculate the S parameters. Since the expansion monitor automatically returns the expansion coefficients for the forward and backward propagating light (a and b), we can calculate the S parameters very straightforwardly. The calculations can be found in the script under the Analysis tab of the "model" group, this script will also export the S parameter results into a .txt file, which can be imported directly by INTERCONNECT.
 As shown in the figures above, the Results View will automatically show the S parameters result returned by the model analysis group. One can then visualize this result by rightclicking on "S" and selecting Visualize.
Monte Carlo analysis
To test how our design is affected by fabrication errors, we can use either a parameter sweep or an Monte Carlo analysis project.
In ring_resonator2_Monte Carlo.lms, a "FSR" analysis group has been added, which will return the FSR by finding the peaks in the transmission spectrum of the "through" monitor. We will track the change in the FSR as a function of the waveguide width and height, assuming a fabrication error of ±10nm.
Parameter sweep
A nested parameter sweep project has been set up to track the change in FSR as a function of the width of the waveguide (from 0.39 to 0.41 microns) and the height of the waveguide (from 0.17 to 0.19 microns). Once the sweep is complete, one can plot the map of the FSR as a function of the waveguide height and width to see how the result deviates from the original design as a result of this ±10nm fabrication error.
Monte Carlo analysis
An Monte Carlo analysis project has also been set up to vary the width of the waveguide based on a Gaussian distribution centered at 0.4 microns, with a standard deviation of 0.01 microns. Once this is run, we will be able to see whether the FSR falls within our target specification range of 27nm to 27.5nm.
Parameter extraction results
The ring resonator is a 4 port device, which we we can label 1 through 4, as shown below. We can use a mode expansion monitor to calculate the complex mode expansion coefficient for both forward and propagating modes in each waveguide. This allows us to easily construct the 16 parameter S matrix which can be exported for use in INTERCONNECT. In reality, this device is so symmetric, that only 4 coefficients of the S matrix need to be calculated  for example, S11=S22=S33=S44.
The mode expansion monitor is setup to calculate the amount of forward and backward propagating power in the fundamental TE mode for the 4 monitors at the input and output ports. First, we can look at this in the Visualizer. Note that this analysis takes several seconds because each waveguide mode is recorded over 500 frequency points. To speed up the calculation, we have used a single mode at the center frequency for the expansion, however we could calculate more mode profiles over the device bandwidth to obtain a more accurate expansion. Once calculated, the expansion is stored in memory and will be saved to the .lms file for quick future reference. The figure below shows the amount of power reflected in port 1 and transmitted through ports 2, 3 and 4 (T forward/T backward).
It is interesting to note the resonant reflection and transmission that is occurring at port 1 and port 4 (blue and green). The power reflected and leaking out port 4 is equivalent. These are due to weak coupling between forward and backward propagating modes in the ring, which can have a substantial effect due to the high Q of the device.
The 'model' itself is an analysis group that is setup to calculate the S parameters. Select the model and use the Results Manager to calculate the S matrix. During the calculation, S11, S21, S31 and S41 are saved to the text file MODEtoINTERCONNECT.txt which can be used to create a ring resonator element in INTERCONNECT. The different S parameters can be easily visualized. For example, below we see the phase of S21 and S31. We can see the effect of the resonances which lead to sudden changes in the slope of the phase which indicates the sudden change in group delay at resonance.
Comparing with 3D FDTD
The same ring resonator is modeled using 3D FDTD in the Ring Resonator FDTD page, and the results are shown below:
These are in reasonable agreement with the Propagator results shown in the previous section (especially in the FSR). There are some differences in the Q factors, which is not too surprising as FDTD accounts for more sources of loss. That being said, one can go a long way towards optimizing the design with only Propagator simulations. Below is a summary of the simulation requirements for the two types of solvers:
Note that this is a relatively small simulation (10x10um span in the x/y directions). Typical simulations with ring resonators or other silicon photonic devices require much larger simulation regions and much longer simulation times. In that case, it is even more important to consider using MODE' Propagator, which may lead to a significant amount of time savings.
Monte Carlo analysis
To make sure that the actual device will work as expected, it is often necessary to consider imperfections that can result from the fabrication process. To do this, we first set up a nested parameter sweep to track change in FSR as a function of the waveguide height and width (assuming fabrication error of ±10nm). The following figure shows the map of the FSR vs waveguide height and width:
Then, in the Monte Carlo analysis, we can define the target range for the FSR. Once the simulations finishes running, the log at the bottom of the "Monte Carlo analysis status" window will show the calculated yield percentage which corresponds to the percentage of trials that fall within the specified yield estimate range. One can also plot the FSR histogram as shown below. (Note that even though we are only considering the FSR in this example, it is very straightforward to extend this analysis to take into account other properties such as the shift in the resonance peaks, the Q factors ... etc using the same methodology.)
The parameter sweep and Monte Carlo analysis shown in this example required more than 100 simulations. More simulations will be necessary if we want to change more parameters. This is another reason to consider using the Propagator instead of running 3D FDTD simulations.
References
Hammer, M. and Hiremath, K.R. and Stoffer, R. (2004) Analytical approaches to the description of optical microresonator devices. (Invited) In: Microresonators as Building Blocks for VLSI Photonics, 1825 October 2003, Erice, Italy. pp. 4871. AIP Conference Proceedings 709. Springer. ISSN 0094243X ISBN 9780735401846