---
title: "Passive Optical Resonator Interactive Lab | Sandbox Physics"
description: "How can two mirrors with no gain between them pick out frequencies and store light? Returning light interferes with the new drive."
canonical_url: "https://sandboxphysics.com/labs/passive-optical-resonator/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/passive-optical-resonator/"
last_modified: "2026-10-28"
---

L05 · Gain and feedback

# Passive Optical Resonator

**How can two mirrors with no gain between them pick out frequencies and store light?** Returning light interferes with the new drive. Frequencies that return in step build the strongest stored field; detuning weakens the buildup. When the drive stops, the stored energy loses a fixed fraction each round trip. Smaller total losses give a longer memory, without any gain.

Sweep through resonance and watch recorded light build up between partially transmitting mirrors. Switch off the drive to compare the decay.

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

The model follows the envelope of a single axial mode, sampled at the input mirror once per round trip. The mirrors scatter without absorbing, and the distributed attenuation is symmetric; reflectivities are power fractions. All powers in the last-step energy balance come from the same time sample. The frequency offset is measured from a reference longitudinal resonance at each chosen length. Counterpropagating channel markers are schematic and do not resolve the optical wavelength. Gain, transverse stability and frequency-dependent mirror dispersion are not included.

Laser foundations · Light, gain and feedback

[L01 · Compare sources](https://sandboxphysics.com/labs/laser-source-comparison/) [L02 · Count transitions](https://sandboxphysics.com/labs/light-matter-transitions/) [L03 · Build inversion](https://sandboxphysics.com/labs/population-inversion/) [L04 · Amplify once](https://sandboxphysics.com/labs/single-pass-amplifier/) [L05 · Add feedback](https://sandboxphysics.com/labs/passive-optical-resonator/) [L06 · Start from noise](https://sandboxphysics.com/labs/laser-noise-startup/) [L07 · Build and debug](https://sandboxphysics.com/labs/build-debug-laser/) [L08 · Measure and diagnose](https://sandboxphysics.com/labs/laser-diagnostic-bench/) [L09 · Select axial modes](https://sandboxphysics.com/labs/gain-longitudinal-modes/) [L10 · Confine the beam](https://sandboxphysics.com/labs/stable-resonator-geometry/) [L11 · Select spatial modes](https://sandboxphysics.com/labs/transverse-laser-modes/) [L12 · Propagate and measure](https://sandboxphysics.com/labs/gaussian-beam-quality/) [L13 · Align and match](https://sandboxphysics.com/labs/alignment-mode-matching/) [L14 · Optimize extraction](https://sandboxphysics.com/labs/output-coupler-optimization/) [L15 · Drive the dynamics](https://sandboxphysics.com/labs/laser-dynamics-relaxation/) [L16 · Store and release](https://sandboxphysics.com/labs/q-switched-giant-pulse/) [L17 · Lock phases into pulses](https://sandboxphysics.com/labs/mode-locking-frequency-comb/) [L18 · Calibrate the comb](https://sandboxphysics.com/labs/optical-frequency-comb/) [L19 · Inject carriers](https://sandboxphysics.com/labs/semiconductor-laser-diode/) [L20 · Compare gain media](https://sandboxphysics.com/labs/laser-medium-zoo/) [L21 · Distributed gain](https://sandboxphysics.com/labs/fiber-fbg-laser/) [L22 · Choose feedback](https://sandboxphysics.com/labs/laser-feedback-architectures/) [L23 · Balance ultrafast pulses](https://sandboxphysics.com/labs/ti-sapphire-oscillator/) [L24 · Stretch and compress](https://sandboxphysics.com/labs/chirped-pulse-amplification/) [L25 · Trap and extract](https://sandboxphysics.com/labs/regenerative-multipass-amplifier/) [L26 · Convert frequencies](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/) [L27 · Scatter and cascade](https://sandboxphysics.com/labs/raman-brillouin-lasers/) [L28 · Synchronize a slave](https://sandboxphysics.com/labs/injection-locking/) [L29 · Lock to a reference](https://sandboxphysics.com/labs/pdh-frequency-lock/) [L30 · Measure phase noise](https://sandboxphysics.com/labs/laser-linewidth-phase-noise/) [L31 · Scatter for feedback](https://sandboxphysics.com/labs/random-laser/) [L32 · Diagnose thermal scaling](https://sandboxphysics.com/labs/thermal-lensing/) [L33 · Combine coherent fields](https://sandboxphysics.com/labs/coherent-beam-combining/) [L34 · Sustain a Kerr pulse](https://sandboxphysics.com/labs/microresonator-kerr-comb/) [L35 · Return a delayed field](https://sandboxphysics.com/labs/external-feedback-chaos/) [L36 · Scan without mode hops](https://sandboxphysics.com/labs/tunable-laser-scan/)

Physical basis: [MIT: coherence and spectrum](https://ocw.mit.edu/courses/res-6-006-video-demonstrations-in-lasers-and-optics-spring-2008/resources/coherence-length-and-source-spectrum/) · [Rate equations](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter2/) · [Pumping and gain media (§7.2)](https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/bf3bdeb971819ddc9c68719d28bb2745_classnotes.pdf) · [Laser dynamics](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter4/)

Physics tutorial

## Add fields before measuring power

**Background** Light that comes back after a round trip has picked up a phase and lost some amplitude. At the input mirror it interferes with the newly injected light. Repeating this trip after trip picks out certain frequencies and lets the cavity remember earlier input.

**Why it matters** Because energy builds up over many round trips, the power circulating inside the cavity can exceed the power the source supplies at any instant.

Start with the essentials

- **Focus question**: If you shift the overall phase of the input, does the steady transmission change, or only the way it gets there?
- **One-sentence intuition**: Phase decides whether each return adds constructively, and loss decides how long the field lasts. Reflected, transmitted and lost power, plus the change in stored energy, always add up to the input.

### Core mathematical model

#### The field after each round trip

$$
a_{n+1}=\sqrt{R_1R_2S}\,e^{i\phi_n}a_n+\sqrt{1-R_1}\,d_n,\qquad\phi_n=2\pi\Delta\nu_n/\mathrm{FSR}
$$

Fixed mirror phases are folded into the reference resonance. S is the fraction of power that survives the distributed loss in one round trip. The input phase is a separate, overall phase of the drive.

#### Steady Airy transmission

$$
\frac{P_t}{P_{\mathrm{in}}}=\frac{(1-R_1)(1-R_2)\sqrt S}{1+r^2-2r\cos\phi},\qquad r=\sqrt{R_1R_2S}
$$

Over half a round trip, the surviving power fraction is the square root of S. The curve is the steady response at a single frequency; a scan at finite speed may not reach it.

#### Mode spacing and photon lifetime

$$
T_R=\frac{2L}{c},\quad\mathrm{FSR}=T_R^{-1},\quad\tau_p=-\frac{T_R}{\ln(R_1R_2S)}
$$

The photon lifetime is the exact time constant of the exponential energy decay, sampled once per round trip. All cavity lengths are vacuum optical lengths.

#### Exact linewidth and impedance matching

$$
\delta\nu=\frac{2\,\mathrm{FSR}}{\pi}\arcsin\frac{1-r}{2\sqrt r},\quad\mathcal F=\frac{\mathrm{FSR}}{\delta\nu},\quad R_1=R_2S
$$

The allowed reflectivity range guarantees that each resonance has a half-maximum point. On resonance, the matching condition cancels the steady reflected power, although internal loss can still keep transmission below one.

#### Energy balance for one round trip

$$
P_{\mathrm{in},n}=P_{r,n}+P_{t,n}+P_{\mathrm{loss},n}+\left(|a_{n+1}|^2-|a_n|^2\right)
$$

Multiplying by the round-trip time turns this power balance into an energy balance for one sampled packet. The stored power is sampled just after the new input arrives; the outgoing powers come from the packet that circulated just before.

### Common difficulties

#### Circulating power is stored energy moving around

**Typical misconception** If the circulating power is large, the cavity must be creating energy.

**Better mental model** The energy balance for each step includes the change in stored energy. Passive mirrors only redirect energy, and internal attenuation removes it.

#### Input phase is not the cavity phase

**Typical misconception** Changing the overall input phase moves the resonance frequency.

**Better mental model** A constant overall phase just rotates the whole steady field. Resonance depends on the phase picked up in one round trip; a sudden jump in input phase causes a transient.

### Run the experiment

1. 01

   #### Build the field one round trip at a time

   Reset to On resonance and replay the recorded buildup. Expand Last encounter to inspect the returned, injected and summed vectors; Advanced controls also offers Record one actual round trip.

   **What to observe:** Each in-phase addition makes the circulating field larger. In steady state, the light reflected straight off the input mirror cancels the light leaking back out of the cavity.

2. 02

   #### Change phase without changing power

   In Advanced controls, shift Input phase by 180 degrees. The new 300-trip acquisition continues the stored field; compare its transient with the endpoint readouts.

   **What to observe:** At first the old stored field interferes with the shifted drive. Once it settles, the transmitted power is back at its earlier value.

3. 03

   #### Measure ring-down

   After buildup, uncheck Drive on. Compare powers taken the same number of round trips apart.

   **What to observe:** The stored power drops by the same survival factor every round trip, because nothing new is injected into the phasor.

4. 04

   #### Impedance matching and frequency scans

   Compare the Impedance matched and Internal loss presets, then try Continuous scan.

   **What to observe:** Matching suppresses the reflected power on resonance. The scan trace includes the finite time the field needs to build up; the Airy plot stays as a labeled steady reference.

Physical conclusion

A passive cavity stores light and picks out frequencies, but it never makes energy: with the drive off, the field always rings down.

## Related experiments and articles

Continue exploring

### Build up, scan, switch off

How are a narrow resonance and a long storage time connected?

1. [You are here **Passive Optical Resonator** Follow successive round trips, then switch off the input.](https://sandboxphysics.com/labs/passive-optical-resonator/)
2. [Lab **Fabry–Pérot Passive Cavity** Compare transmission, reflection, linewidth and ringdown.](https://sandboxphysics.com/labs/fabry-perot-passive-cavity/)

Probes and measurement · Reading path

### Controlled light and precision measurement

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How an ammonia beam became a microwave oscillator**](https://sandboxphysics.com/stories/probe-evidence/maser-inversion-and-feedback/)
2. [Lab **Absorption, Spontaneous & Stimulated Emission** Watch a probe beam grow, then fade, in a prepared gas](https://sandboxphysics.com/labs/absorption-spontaneous-stimulated-emission/)
3. [Lab **Pumping & Population Inversion** Pump three gases with one lamp and see which one amplifies](https://sandboxphysics.com/labs/pumping-population-inversion/)
4. [Lab **Stable Resonator & Transverse Modes** Curve two mirrors and size an iris until one beam shape is left](https://sandboxphysics.com/labs/stable-resonator-transverse-modes/)
5. [Lab **Build & Debug Your First Laser** Test gain against cavity loss](https://sandboxphysics.com/labs/build-debug-laser/)
6. [Lab **Laser Noise Startup Workbench** Follow a cavity mode from its first seed](https://sandboxphysics.com/labs/laser-noise-startup/)
7. [Lab **Laser Startup from Noise** Watch many cavity frequencies compete for one gain](https://sandboxphysics.com/labs/laser-startup-from-noise/)

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)