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, we show how MODE can be used to design a ring resonator. The user will learn to:
 Insert a ring resonator object from the components library
 Use the Eigenmode Solver to choose the waveguide spacing, coupling length and ring length for the desired FSR and Q factor
 Compare results with the theoretical design and 3D FDTD results
This page contains 6 sections. The first section describes how to setup the model. You can skip this step, open the associated .lms file and proceed to the following steps if you want to know the results first. Section 2 and section 3 use the eigenmode solver and the analytical results from the discussion and results section to design the ring resonator. The fourth and sixth sections discuss how to set up the propagator simulation. The fifth section shows how to compare the propagator results with analytical and 3D FDTD results.
Create ring resonator
 Begin by starting MODE. You can save the MODE Simulation Project file (.lms) at any point in this process. To do so, choose SAVE in the FILE menu.
 Press on arrow on the STRUCTURES button and select a RECTANGLE from the pulldown menu. Set the properties of the insulator substrate rectangle according to the following table.
tab 
property 
value 

Geometry 
x (μm) 
0 
x span (μm) 
16 

y (μm) 
0 

y span (μm) 
16 

z (μm) 
2 

z span (μm) 
4 

Material 
material 
SiO2 (Glass)  Palik 
 Press on arrow on the COMPONENTS button and select INTEGRATED OPTICS from the pulldown menu. This will open the object library window.
 Select RING RESONATOR from the list and press the INSERT button.
 Set the properties of the ring resonator according to the following table. The coupling length and radius used for the first part of the simulation are just an initial guess and will be modified to the correct values later.
tab 
property 
value 

Properties 
x, y, z (μm) 
7,0,0.09 
Lc (μm) 
0 

gap (μm) 
0.1 

radius (μm) 
4 

material 
Si (Silicon)  Palik 

base width (μm) 
0.4 

base height (μm) 
0.18 

x span (μm) 
14 

base angle 
90 
Add eigenmode solver and find group index
 Press on the arrow on the on the SIMULATION button and select the EIGENSOLVER from the pulldown menu. Set the properties according to the following table. This will place the eigensolver region at the input bus for the ring resonator.
tab 
property 
value 

General 
solver type 
2D X normal 
Geometry 
x (μm) 
5 
y (μm) 
3.6 

y span (μm) 
4 

z (μm) 
0.075 

z span (μm) 
1 

Mesh settings 
mesh cells z 
100 
mesh cells y 
200 
 Press on the RUN button to open the eigenmode solver analysis window. Press on the MESH STRUCTURE button to plot the meshed structure. Set the wavelength to 1.5 microns.
 Click the "Calculate Modes" button and then, select the mode of interest (mode #1) in the Mode list
 Switch to the frequency analysis tab and set the following parameters:
property 
value 

track selected mode 
yes 
stop wavelength (um) 
1.6 
number of points 
4 
detailed dispersion calculation 
yes 
 Click on the FREQUENCY SWEEP button to begin the scan. The scan will take about a minute.
 To plot the calculated dispersion as a function of wavelength, select the FREQUENCY PLOT tab in the bottom righthand corner of the frequency analysis window. Then select "group index" in the plot pull down menu. The plot can be seen above the frequency plot tab. If you press the PLOT IN NEW WINDOW you will get a new window. If you zoom into the plot, you can see that at 1.55 microns, the group index is about 4.63.
Determine coupling length
 Press the SWITCH TO LAYOUT button . This deletes all the mode data and allows us to edit the eigenmode solver.
 Move the center of mode solver to x = 0 so that the mode solver is located in a region where two waveguides cross the mode solver.
 Press on the RUN button to open the eigenmode solver analysis window. Press on the MESH STRUCTURE button to plot the meshed structure. Set the wavelength to 1.55 microns, and press FIND MODES.
 The neff values for the first and second modes are given in the mode list. We can plug in these neff values into an analytical formula (see the discussion and results page) in order to determine the coupling length.
Update ring resonator properties and run propagator simulation
 Press the SWITCH TO LAYOUT button so that it is possible to edit the simulation.
 Set the properties of the ring resonator using the values of the coupling length and the radius determined from the previous steps, ie
tab 
property 
value 

Setup à Variables 
radius (μm) 
3.1 
 Press the arrow on the on the SIMULATION button and select the varFDTD from the pulldown menu.
 Set the properties according to the following table.
tab 
property 
value 

General 
simulation time (fs) 
5000 
Geometry 
x (μm) 
0 
x span (μm) 
10 

y (μm) 
0 

y span (μm) 
10 

z (μm) 
0 

z span (μm) 
1 

Effective index 
bandwidth 
broadband 
 There is a green cross in the graphical user interface which sets the location of the core slab mode for the propagator. Click on PROP object to select the cross and then drag it so that it overlaps with the Silicon portion of the ring resonator.
 Since the effective index is a broadband material, the propagator finds a material fit to the effective index data before running the simulation. Press on the arrow next to the CHECK button and select the MATERIAL EXPLORER. This will open the material explorer, where you can check the fit to the slab mode at the location you just selected. You can also see the material fit for the test materials (which are shown by blue crosses in the graphical user interface).
 Next, reedit the propagator region and select the EFFECTIVE INDEX tab. The plot of the slab mode should look like the image below.
 Press the arrow on the on the SOURCES button and select the MODE source from the pulldown menu. Set the properties according to the following table.
tab 
property 
value 

Geometry 
x (μm) 
4.5 
y (μm) 
3.6 

y span (μm) 
3 

Frequency/Wavelength 
wavelength start (μm) 
1.5 
wavelength stop (μm) 
1.6 
 In the previous step we calculated the coupling length from the effective index difference from the mode solver. You can see the effective index difference which the propagator sees by setting x = 0 so that the mode source is in between two waveguides. Then choose to user select the modes. This technique is shown in a bit more detail in the waveguide couplers application example in the MODE online help.
Add monitors for the propagator simulation
 Press on the arrow on the on the Monitors button and select the frequency domain field and power monitor from the pulldown menu. Set the properties according to the following table
tab 
property 
value 

name 
drop 

Geometry 
monitor type 
Linear Y 
x (μm) 
4.2 

y (μm) 
3.6 

y span (μm) 
2 

General 
override global monitor settings 
yes 
frequency points 
500 
 Press the DUPLICATE button to create a copy of the monitor. Set the properties according to the following table.
tab 
property 
value 

name 
through 

Geometry 
x (μm) 
4.2 
y (μm) 
3.6 
 Press on the arrow on the on the Monitors button and select the field time monitor from the pulldown menu. Set the properties according to the following table
tab 
property 
value 

name 
time_drop 

Geometry 
x (μm) 
4.2 
y (μm) 
3.6 
 Press the DUPLICATE button to create a copy of the monitor. Set the properties according to the following table
tab 
property 
value 

name 
time_through 

Geometry 
x (μm) 
4.2 
y (μm) 
3.6 
Run simulation and plot results
 Press on the RUN button to run the propagator simulation. The job manager will appear and show the progress for the simulation.
 Once the simulation finishes running, all the monitors and analysis groups in the object tree will be populated with data. The Results View window (which can be opened by clicking on the "Show result view" button) will display all the results and their corresponding dimensions/values for the selected object. Plot the time signal and spectrum Ey by rightclicking on the "time_drop" time monitor and selecting Visualize  E or spectrum. (The field profiles can also be visualized in the same way.)
Compare dropped optical power with 3D FDTD and theory
The files which are mentioned in this section can be found in the Examples subdirectory of the installation directory, or downloaded from the online MODE Knowledge base.
 Open the script file editor.
 Press on the OPEN button and browse to the ring_resonator.lsf script file.
 Save the fdtd_results.ldf file in the same folder. This lumerical data file (*.ldf) contains the results from the 3D FDTD simulation. This data file can also be created by running the ring_resonator.fsp 3D FDTD simuluation followed by the ring_resonator_fdtd.lsf script file. The aforementioned script file will automatically generate the data file.
 Press on the RUN SCRIPT button The script will calculate the theory and send a data set to the visualizer. The data set contains the analytical, the 3D FDTD and the propagator transmission through the drop channel (or output bus).
Add other monitors to the propagator simulation
The ring_resonator.lms simulation which is included in the Examples subdirectory of the installation directory, or can be downloaded from the online MODE Knowledge base contains three additional monitors which have not been included in the above instructions.
The three additional monitors are
 a field profile monitor
 a movie monitor
 an effective index monitor
Field profile monitor
The field profile can be created similar to the add and drop monitors which were created earlier. The field profile monitor in this example was set up after the simulation ran to completion once. That way, we can only set the monitor to record data exactly at a frequency which corresponds to the maximum power dropped through the drop monitor.
Movie monitor

Effective index monitor
The effective index monitor is also created similar to the other monitors in the simulation. It shows how the simulation volume is compressed in the z dimension. In contrast, an index monitor will only plot the refractive index of the structure. Further details can be found in the Effective index monitor  Simulation object page.
The most basic configuration of the microring resonator is shown in the image below. It consists of a ringshaped waveguide coupled to two optical waveguides. The cavity mode is excited by evanescent coupling between closely spaced optical waveguides.
The major defining quantities are the average (effective) ring length L, the complex propagation constant β of the circulating mode, the field transmission coefficients of the waveguide coupler t_{11} and t_{12}, and the bend loss.
The real part of the propagation constant β is the phase constant. The total ring loss is due to the imaginary part of beta, the bend loss, and other scattering losses at the coupling region. For the high index contrast waveguides we are considering, at wavelengths around 1550nm, the propagation loss and bend loss are small. For this analysis we will neglect all losses, but the different loss contributions could easily be considered in a more detailed analysis.
For the case of the ringshaped waveguide coupled to two optical waveguides, the dropped optical power can be expressed as
$$P_{D}=P_{I N} \frac{\left\tau_{12}\right^{4}}{\left1\tau_{11}^{2} e^{\imath \beta L}\right^{2}}$$
On resonance, the phase acquired by the wave after a complete roundtrip is an integer multiple of 2π, i.e., βL = 2πN, with N as the mode number. If the effective refractive index is independent of wavelength, then ring length (also called the perimeter) is an integer multiple of the effective wavelength on resonance, or L = Nλ0/neff, where the effective refractive index is defined as neff = cβ/ω and λ0 = c/f0 is the freespace wavelength at the resonance frequency f0.
In reality, the effective index does depend on wavelength and the resonances are separated by the free spectral range (FSR) which, in wavelength units, is given by
$$F S R=\frac{\lambda^{2}}{n_{g} L}$$
where n_{g} = c(dβ/dω) is the group index.
Since the bending loss is low for high index waveguides, and we are ignoring other sources of loss, the Q factor is approximately given by
$$Q=\frac{\lambda}{2 \delta \lambda}=\frac{n_{g} L \pi}{\lambda} \frac{\left\tau_{11}\right}{1\left\tau_{11}\right^{2}}$$
We will now use the above formulas to design a ring resonator for a WDM system with a channel spacing of 200GHz (1.6nm at 1550nm). We want to design the system to drop every 16th channel. The FSR should therefore be 3200GHz (25.6nm at 1550nm). We would like the FWHM of the drop to be 100GHz, corresponding to a Q of approximately 1550nm/0.8nm ~ 2000.
We will use a system made of an SOI waveguide. The waveguide is 400nm wide and 180nm high.
Ring resonator design
Step 1: Find total length of the ring needed to obtain desired FSR
We use the Eigenmode Solver on the 3D waveguide cross section to calculate the group index from 1500nm to 1600nm. We see that the group index is approximately 4.63 at 1550nm. This is significantly larger than the effective index at this wavelength.
Now that we know the group index, we can easily calculate the total desired ring length.
$$L=\frac{c}{n_{g} F S R} \approx 20.2 \mu m$$
Step 2: Find the length and gap of the coupler segment
We want to design a resonator with a Q factor of 1.55mm/0.8nm ~ 2000. We can solve for the amplitude of t for a given Q with
$$\left\tau_{11}\right=\sqrt{\left(\frac{n_{g} L \pi}{2 Q \lambda}\right)^{2}+1}\frac{n_{g} L \pi}{2 Q \lambda}$$
Using our known group index, ring length and center wavelength, we find t_{11} ~ 0.95 and therefore t_{12} ~ 0.31
If we use the Eigenmode Solver to calculate the index of the coupled waveguide system, we know that the coupling length can be determined from the difference in effective index of the symmetric and antisymmetric coupled modes by the formula
$$L_{\text {coupler}}=\frac{\lambda}{\pi \Delta n} \sin ^{1}\left(\left\tau_{12}\right\right)$$
Symmetric mode (Ey) 
AntiSymmetric mode (Ey) 
Using a 100nm gap between the waveguides, we find deltan = 0.109 at 1550nm. This gives a coupling length of approximately 1427nm. In reality, we will use a coupling length of 0 and will obtain enough coupling from the bent section of the ring near the straight waveguide.
The radius of the ring can be chosen such that we have the desired coupling length and the desired total ring length. For the remainder of the example, we will use a radius of 3.1 microns, and we will use this design for the Propagator simulation in the next step.
Propagator simulation
In this Propagator simulation, the MODE source will calculate the fundamental TE mode of the waveguide,and use this to inject a guided mode into the upper waveguide. For an overview of the 2.5D variational FDTD solver, see the Lumerical’s 2.5D FDTD Propagation Method whitepaper on our website.
The ring resonator is a high Q device which traps the light for many round trips in the ring. These high Q devices require longer simulation times in the time domain than nonresonant devices. We will start with a simulation time of 5000fs, although more time may be necessary. Note that this is longer than our default simulation time (1000fs). It is important to increase the simulation time because the frequency domain monitor results are incorrect if the simulation time is not set long enough for the fields to decay.
After running the simulation, we can consider the E field intensity near a drop resonance.
Results
We can compare the result with the theoretical curve based on our target FSR and Q. Note that we have adjusted the phase of the coupling coefficient to align the peaks near 1550nm, since only the amplitude of the coupling coefficient is given by Q.
The results are in reasonable agreement for the theoretical FSR but the Q factor is incorrect. This is due to
 neglecting all sources of loss in the theoretical curve calculation
 not achieving exactly the desired value of t_{12}
The 3D FDTD simulation shows lower total transmission because it accounts for more sources of loss.
Note that we have adjusted the theoretical peaks to give a maximum at 1550nm. The precise position of this peak is very sensitive to the exact optical length of the ring.
Next Steps
Please see Part 2 of the ring resonator tutorial: Ring resonator (parameter extraction and Monte Carlo analysis), where we will consider how to carry out the parameter extraction and Monte Carlo analysis process for this design.
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