Optical ring resonators are key components in the field of integrated photonics. The unique capability of ring resonators to selectively interact with specific wavelengths of light makes them extremely versatile. They can be utilized in a broad array of applications, including optical filtering, modulating, switching, and sensing. However, simulating a ring resonator can be computationally expensive due to the high-Q resonances. Alternatively, we can investigate the coupling between a straight waveguide to a ring only by simulating the coupling region. The coupling coefficients can be extracted from this much simpler simulation.

Noticeably, in a waveguide-to-ring simulation, one of the simulation domain boundaries intersects the ring structure. Structures that are not translationally invariant inside a PML can make an FDTD simulation diverge. Tidy3D now enables extrude_structures=True by default for PML, so near-boundary structures are automatically extended through the PML to enforce translational invariance. This default avoids the divergence shown in older versions of this notebook, but it also changes the effective geometry inside the absorbing layer.

In this notebook, we first disable structure extrusion on the PML boundary that intersects the ring to reproduce the divergence mechanism. We then discuss the current default PML behavior and use an Absorber boundary as a conservative alternative for this curved geometry. The absorber functions similarly to PML by absorbing outgoing radiation to mimic infinite space. In standard (translationally invariant) setups, the absorber usually has slightly higher reflection and requires more computation than PML. However, it is numerically robust for slanted geometries, and does not require changing the structure into an extruded continuation inside the absorbing layer, which in itself can introduce numerical reflections.

FDTD simulations can diverge due to various reasons. If you run into any simulation divergence issues, please follow the steps outlined in our troubleshooting guide to resolve it. The same guide also discusses PML structure extrusion and its possible reflection artifacts in a simpler slanted-waveguide example.

For more integrated photonic examples such as the 8-Channel mode and polarization de-multiplexer, the broadband bi-level taper polarization rotator-splitter, and the broadband directional coupler, please visit our examples page.

If you are new to the finite-difference time-domain (FDTD) method, we highly recommend going through our FDTD101 tutorials.

Simulation Setup¶

Define simulation wavelength range to be 1.5 $\mu m$ to 1.6 $\mu m$. .

The ring and waveguide are made of silicon. The top and bottom claddings are made of silicon oxide. Here we use the materials from Tidy3D's material library directly.

Define the geometric parameters. The waveguide is 500 nm wide and 220 nm thick. The ring has a radius of 5 $\mu m$ and the gap size is 50 nm.

We only need to define two structures: a straight waveguide and a ring. Both are commonly used PIC components introduced in the demonstration notebook. We can simply copy the associated functions here and them use to define the structures quickly. Namely, we copy the straight_waveguide function and the ring_resonator function.

Use the above functions to define the Structures.

We will use a ModeSource to excite the straight waveguide using the fundamental TE mode. A ModeMonitor is placed at the through to measure the transmission. Another ModeMonitor is placed at the ring to measure the coupling to the ring. For the monitor at the ring, we need to properly set up angle_theta, angle_theta, bend_radius, and bend_axis in the ModeSpec as demonstrated in the tutorial. Finally, we add a FieldMonitor to help visualize the field distribution.

PML Boundary with Structure Extrusion Disabled¶

First, we define the simulation with a PML on each side, but explicitly set extrude_structures=False on the positive $y$ boundary where the ring intersects the simulation domain boundary. This reproduces the unstable setup that would have been obtained from the old PML default.

Submit the simulation to the server to run.

Now we see that the status of the job is shown to be diverged. This is expected because we intentionally disabled structure extrusion on the positive $y$ PML boundary. The ring section in that PML layer is not perpendicular to the boundary and is not translationally invariant, as shown in the zoomed-in plot below. We are also given a warning about the divergence and prompted to check the log for more information.

The log tells us that there are two potential causes of divergence in this simulation: the presence of dispersive materials in the PML, and the fact that the ring is not translationally invariant in the PML region. While both can cause divergence, our built-in silicon material model has been fit to perform well for straight waveguides entering the PML at telecommunication wavelengths. Thus, the more likely cause of the divergence is the ring geometry that is not translationally invariant in the PML when extrusion is disabled.

This can be further confirmed by plotting the electric field distribution. From the plot, we can see where the field built up and caused the simulation to diverge. As expected, the field built up around the ring and PML interface.

Default PML Boundary with Structure Extrusion¶

With the current default td.PML(), extrude_structures=True. If a structure is close enough to a PML boundary, Tidy3D automatically extends it through the PML along the direction normal to the boundary. This makes the material profile translationally invariant in the PML and prevents the divergence shown above.

This default is often the right choice when the structure physically should continue out of the simulation domain, such as a straight waveguide leaving the domain. However, it is not always a physically neutral operation. In this example, the original boundary intersects a curved ring; extrusion replaces the part of that curve inside the PML with an artificial continuation sampled from the material near the boundary. That artificial continuation can create unphysical reflections or phase errors even though the simulation remains numerically stable.

The runtime log is useful for detecting this situation: Tidy3D will warn that structure extrusion through PML boundaries overwrote original translational variance, which means the original geometry inside PML was not translationally invariant but extrusion made it invariant. We will run this default PML case and compare its reflected power against the absorber case below.

Submit the default PML simulation. This run should remain stable because the positive $y$ PML now extrudes the ring structure through the absorbing layer.

The simulation finishes, but the log should still be inspected. A warning that structure extrusion overwrote translational variance is a sign that the stable PML result may include an artifact caused by the geometry change.

Using Absorber Boundary¶

For this geometry, switching the positive $y$ boundary to Absorber is a conservative remedy. The absorber avoids the PML divergence without replacing the curved ring segment in the absorbing region by an extruded continuation. Since the ring only intersects the positive $y$ boundary, we only need to apply the absorber there and can keep PML on the other sides.

In addition, we increased the number of layers in the Absorber to 60 to ensure sufficient absorption and minimal reflection.

After switching to Absorber, the simulation does not run into the divergence issue anymore. We compare this absorber result with the default PML result to see whether automatic extrusion introduces a measurable artifact. The usual forward through and drop powers are a useful first check, but a more sensitive diagnostic is the power in the opposite direction at the ring drop monitor.

The forward through and drop spectra are nearly unchanged by the boundary choice. This means the main coupling quantities that this simulation with a truncated domain is designed to extract are not strongly affected by the PML extrusion in this setup. To look for a more subtle artifact from boundary reflection, we can use the drop monitor to measure the mode propagating in the opposite ring direction by selecting direction="-". In an ideal unidirectional coupling picture with no reflection caused by the boundary, this backward component should be zero.

The backward ring power is somewhat larger with PML extrusion than with the absorber. Its peak value is approximately 2.1e-5 for PML extrusion and 1.2e-5 for the absorber, compared with a forward drop power of about 1.6e-1. Thus this monitor detects a small difference between the boundary treatments that is not visible in the forward through and drop spectra. The effect is still very weak in absolute terms, but it is the clearest signature in this notebook of the artificial PML extrusion causing a small reflected ring component.

Finally, we visualize the field distribution for both stable simulations. The two interior fields are very similar for this setup. The absorber result remains the conservative choice when preserving the original curved boundary geometry is more important than the lower nominal reflection of PML, but the spectra above show that the PML extrusion artifact is limited to a weak backward ring component for this particular truncation.

Absorber and PML Extrusion Tradeoffs¶

The best absorbing boundary depends on what should physically happen at the truncated edge of the simulation. Starting from Tidy3D version 2.11, the PML defaults with extrude_structures=True. This is usually preferable when the near-boundary structure should continue straight through the boundary, because it keeps the low reflection of PML while preventing material profiles inside the PML that are not translationally invariant. It is also the default, so many ordinary waveguide terminations work without extra boundary tuning. For a more direct slanted-waveguide illustration of the same tradeoff, see the diverged FDTD troubleshooting notebook.

Absorber is usually preferable when a structure intersects the boundary in a way that should not be automatically continued, as in this curved ring example. It preserves the original geometry at the cost of a thicker absorbing region and typically higher reflection than PML. In this particular simulation, the forward through/drop spectra and bus reflection from default PML extrusion and absorber are nearly indistinguishable. The backward drop monitor is more sensitive and shows a small excess ring-backward component for PML extrusion, but the magnitude is still too weak to change the extracted coupling spectra appreciably. That means the PML extrusion warning should be treated as a prompt to check the observables that matter for the model, not as proof that the result is inaccurate.

For absorbers, there are two possible sources of reflection. The first, and more common one, is that the ramping up of the conductivity is not sufficiently slow, which can be remedied by increasing the number of absorber layers (40 by default). The second one is that the absorption is not high enough, such that the light reaches the PEC boundary at the end of the absorber, travels back through it, and is still not fully attenuated before re-entering the simulation region. If this is the case, increasing the maximum conductivity (see the API reference) can help. In both cases, changing the order of the scaling of the conductivity (sigma_order) can also have an effect, but this is a more advanced setting that we typically do not recommend modifying.