Analyzing an Optical Resonant Structure Using an Eigenfrequency Study
You can simulate an optical resonant structure using the COMSOL Multiphysics® software and its add-on Wave Optics Module. Apart from using an external excitation, you can also directly compute the resonance frequencies, spatial modes, and additional properties, such as the optical quality factor, using an eigenfrequency analysis. Here, we provide an introduction to this approach. After computing the model, we present different options for how to filter the results so you can examine specific modes of interest.
Tutorial: Using Direct Computation to Analyze an Optical Resonant Structure
Follow along in the software as we use an example model of a Fabry–Pérot resonator to show how you can perform an eigenfrequency analysis to compute optical properties.
In the video, we show a step-by-step modeling demonstration that involves adding multiple eigenfrequency studies and changing the settings in each, highlighting different options you can use for refining the results. The options shown include the:
- Eigenfrequency study including all modes
- Eigenfrequency study with filtered modes
- Eigenfrequency study with region search
Details that expand on the software demonstration are included following the tutorial.
Setting Up the Eigenfrequency Analysis to Directly Compute the Resonator Modes
Using an external excitation, you can determine the optical properties of a resonator. Instead of probing the resonator externally with a spatial field distribution at a defined frequency, you can also use an Eigenfrequency study to directly compute the eigenmode, resonance frequency, and losses in the resonant structure. Specifically, we solve for the mode field and the complex eigenvalue. From this eigenvalue (
), you can derive the quantities of interest, like resonance frequency, the quality factor, and damping.
A simple resonant structure with a stable spatial mode is a Fabry–Pérot resonator, hence why we have chosen this type of device for our model example.
We will set the model up as two spherical boundaries that are defined using a Perfect Electric Conductor boundary condition to represent two perfect mirrors. We can reduce the number of degrees of freedom and only compute half of the geometry by analyzing the z-component of the field and defining a Perfect Magnetic Conductor boundary condition at the lower boundary. On the top side, we can use a Scattering Boundary Condition to model the open cavity.
A grid with the resonator geometry, shown in gray with a black outline.
The model geometry for the resonator.
As an input for the Eigenfrequency study, we need to provide a desired number of modes and a location in the complex plane where we want to search for the eigenfrequencies.
A screenshot of the Model Builder and the Eigenfrequency study settings window.
The settings for the first Eigenfrequency study.
Twenty eigenmodes of the resonator are shown in a 4 x 5 grid in the prism color table.
Eigenmodes of the resonator. Notice the substantial number of spurious modes that extend to the upper open boundary of the system.
Analyzing Modes
When we compute an Eigenfrequency study, we typically find a large number of modes. Often, a substantial number of the modes that exist mathematically are not truly of interest, as you would not typically excite them with a “normal” optical beam, or they only exist due to a specific combination of boundary conditions. To focus on specific modes of interest, we can search for a larger number of modes and filter the result based on user-defined criteria.
Coming back to the example of a Fabry–Pérot resonator with perfect mirrors, we expect the modes to have a low damping time and thus a high Q factor. We can use this information to remove unwanted, spurious modes. The option to filter the modes is available in the Eigenfrequency study step in the Filtering and Sorting section. In this example, we will only include modes that have a Q factor larger than 1e7 by adding comp1.ewfd.Qfactor>1e7 to the filter expression.
10 filtered outputs of the resonator displayed in 3 rows. The first row has 2 modes, and the second and third row contain 4 modes.
Filtered output of the modes. The modes with a high Q factor are localized more along the optical axis and do not extend to the upper boundary.
The eigenmodes and filtered modes are plotted with all modes depicted with blue circle outlines, and the filtered modes are plotted with green squares. There is a dashed line at 10^7 to indicate the Q factor threshold.
The filter option in the study settings is used to restrict the output to only include modes with a high Q factor.
A suitable filter criterion can depend greatly on the specific situation and can take many forms. Besides filtering by Q factor, a spatial filtering can also be very efficient, which is demonstrated in the Whispering Gallery Mode Resonator tutorial model.
As another option, in the Filtering and Sorting section of the Eigenfrequency study step settings, you can change the order of eigenfrequencies in the output dataset, that is, sort the output. Using the User defined option for the sorting method, you can, for example, use comp1.ewfd.Qfactor to sort the output eigenfrequencies in ascending or descending order by Q factor.
For parametric sweeps, mode following can also be useful to track mode crossings properly, which is discussed in the article How To Track and Categorize Eigenmodes in Parameter Sweeps.
The Model Builder with the Eigenfrequency study selected and the corresponding Setting window showing the Filtering and Sorting section settings, with the Predefined sorting method setting highlighted.
A filtered eigenfrequency study. The Filtering and Sorting section in the Settings window is expanded, along with the options for the Sorting method feature.
Navigating the Complex Plane
In the Eigenfrequency study step, while you can search at a specific frequency, you also have the option to define a search region by switching the Eigenfrequency search method to Rectangle.
The UI showing the Eigenfrequency study step selected in the Model Builder, the corresponding Settings window, and the convergence plot.
The region search option in the Eigenfrequency study step, which can be used to define upper and lower bounds for the real and imaginary parts of the eigenvalue, respectively.
To briefly summarize, each eigenvalue 
corresponds to a point on a complex plane. We can define a complex frequency as 
. The region search enables you to define a rectangle in this complex frequency plane. The real part corresponds to the resonance frequency, 
, while the imaginary part corresponds to the damping in time, 
. With this approach, you can restrict the search region early on and perform a more relevant search as opposed to filtering and removing solutions later. However, the region search can be slower than the default Around shift option, so searching for more modes and removing them using a filter can be quicker.
The Resonance Frequency vs. Damping in time graph, where the low and high Q modes are plotted with blue circle outlines, the high Q modes are plotted with green squares, the region search is plotted with red triangles, and a red dashed line outlines the search region.
Comparison of a study including all modes, filtered modes, and the result of a region search. Searching for modes without a filter often gives results of undesired spurious modes with high damping. The total number of modes computed depends on the settings of the Eigenfrequency study step.
Benefits and Challenges of this Approach
Using this approach enables you to directly compute the eigenmode with its characteristic properties, like frequency, damping, and the mode shape. This functionality can be especially useful for high Q modes that are very narrow in line width, which would be difficult to find and spectrally resolve with a frequency-domain sweep. A challenge is that a very large number of modes can exist mathematically that don’t necessarily all have physical significance, especially for larger systems. Eigenfrequency analysis often involves the inspection of a large number of modes, with only a few being relevant, requiring careful filtering to isolate the desired ones. The best filtering strategy will depend on the specific situation in the model and the mode shape of the resonator. Using the region search option can be helpful if you already know the properties of the mode quite accurately.
Further Learning
- Learning Center content:
- Tutorial models:
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