---
title: "Microring & Whispering-Gallery Resonator Interactive Lab | Sandbox Physics"
description: "Why does a small ring beside a waveguide make the transmitted light dip at certain wavelengths?"
canonical_url: "https://sandboxphysics.com/labs/microring-whispering-gallery-resonator/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/microring-whispering-gallery-resonator/"
last_modified: "2026-10-27"
---

Optics 086 · Waveguides, structured light, and modern optics

# Microring & Whispering-Gallery Resonator

**Why does a small ring beside a waveguide make the transmitted light dip at certain wavelengths?** Light coupled in from the straight waveguide circles the ring and resonates whenever a whole number of wavelengths fits the effective circumference, so a smaller ring has its resonances farther apart. On resonance the ring builds up a strong circulating field. When the coupling to the waveguide equals the ring's own loss, transmission at resonance drops to zero; this is critical coupling.

Couple a ring resonator to a single bus waveguide and follow the light coherently. Set the radius and gap, then read the lossless coupling, the exact resonance width, the circulating power and the shift caused by index changes, and freeze complex spectra.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The ring carries one travelling mode, and the coupler is a lumped element. The gap dependence uses an explicit, assumed fit, and the propagation constant has linear dispersion. The whispering-gallery mode is not solved in 3D, the material is not measured, and the model does not predict fabrication results. Backscattering, thermal and nonlinear effects are left out.

Physics tutorial

## Audit the physics of Microring & Whispering-Gallery Resonator

**Background** Light in the ring keeps coming back to the same coupler. Adding up the round trips as a geometric series gives $H=\frac{t-ae^{i\phi}}{1-tae^{i\phi}}$. The radius sets the circumference, and the group delay sets the spacing between resonances.

**Why it matters** Why does a small ring beside a waveguide make the transmitted light dip at certain wavelengths?

Start with the essentials

- **Focus question**: Why does a small ring beside a waveguide make the transmitted light dip at certain wavelengths?
- **One-sentence intuition**: The ring can be bright while the output is dark. The energy balance is $T+(1-a^2)B=1$. The power circulating in the ring is not the power missing from the output.

### Core mathematical model

#### A lossless coupler

$$
\begin{pmatrix}E_{\rm out}\\E_+\end{pmatrix}=\begin{pmatrix}t&i\kappa\\i\kappa&t\end{pmatrix}\begin{pmatrix}1\\ae^{i\phi}E_+\end{pmatrix},\quad t^2+\kappa^2=1
$$

The fields are normalized so that their squared magnitudes give mode power relative to the input. The ring lies in the X–Z plane, and the forward wave leaves the coupling point at the bottom heading toward positive X. The time dependence uses a negative frequency exponential. There is one bus waveguide and no drop port.

#### Power buildup and loss in the ring

$$
B=\frac{\kappa^2}{|1-tae^{i\phi}|^2},\quad P(s)/P_0=B a^{2s/L},\quad a=10^{-\alpha_{\rm dB}L_{\rm cm}/20}
$$

The buildup factor is the circulating power just after the coupler. Uniform attenuation removes a fraction of that power on every round trip. At resonance, critical coupling happens when the self-coupling amplitude equals the round-trip amplitude. With no intrinsic loss, the steady-state through amplitude stays at one.

#### The assumed dispersion and gap model

$$
\beta(\omega)=\frac{(n_0+\Delta n)\omega_0+n_g(\omega-\omega_0)}{c},\quad K(g)=0.12e^{-(g-0.2\,\mu\mathrm m)/(0.12\,\mu\mathrm m)}
$$

The reference wavelength is 1550 nm, with phase index 2.4 and group index 4.2; these are teaching assumptions. The gap law is an imposed coupler calibration over a limited range, not the result of a Maxwell overlap calculation or a manufacturer curve. The index shift is a shift of the effective mode index at the reference wavelength, not a calibrated refractometer reading of the environment.

#### Exact linewidth and the energy lifetime from round-trip decay

$$
\mathrm{FSR}=\frac1{T_r},\quad\Delta f=\frac{2}{\pi T_r}\arcsin\!\frac{1-at}{2\sqrt{at}},\quad\tau_U=-\frac{T_r}{2\ln(at)},\quad T_r=\frac{n_gL}{c}
$$

The width is the full width at half maximum of the circulating power. The spectral quality factor is the resonance frequency divided by that exact width, and the decay quality factor is the angular frequency times the energy lifetime. The two agree for a narrow resonance, but not exactly for arbitrary loss. After the drive is switched off, the decay plot samples the power once per complete round trip.

### Common difficulties

#### Does zero transmission mean zero field?

**Typical misconception** The ring holds only the power that is missing from the output.

**Better mental model** No. Coherent feedback can build up circulating power far larger than the input. At critical coupling the outgoing amplitudes cancel, and the intrinsic loss dissipates the incoming energy. With no intrinsic loss there is no absorption dip.

### Run the experiment

1. 01

   #### Scene 1: Coherent feedback

   Tune the laser to resonance, then set the intrinsic loss to match the coupler. Look at how the complex fields cancel, and how the input power divides between output and loss on a linear scale.

   **What to observe:** A larger radius gives a smaller free spectral range, and the nearest resonance can switch to a different integer order. The index-shift comparison stays on the same unperturbed order.

2. 02

   #### Scene 2: Lossless all-pass

   Select the lossless all-pass ring and tune to resonance again. The circulating power grows, but the through power stays at one. This is not the absorption dip of critical coupling.

   **What to observe:** The colors in the apparatus show the averaged travelling power on a fixed display scale. They are not photons, they do not resolve the radial mode, and the animation does not run at the real speed.

3. 03

   #### Scene 3: Effective-index perturbation

   Acquire a spectrum, change the effective index, and keep the frozen record as a reference. Then change the radius and look at the real gap in the close-up view.

   **What to observe:** The saved scan holds simulated complex detector data, and the plotted power is recomputed from those same samples.

Physical conclusion

Match the amplitudes, keep track of where the power goes, and do not confuse light stored in the ring with light missing from the output.

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)