---
title: "Stable Resonator & Transverse Modes Interactive Lab | Sandbox Physics"
description: "How should two mirrors be curved and spaced to make a stable cavity that favors the fundamental mode?"
canonical_url: "https://sandboxphysics.com/labs/stable-resonator-transverse-modes/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/stable-resonator-transverse-modes/"
last_modified: "2026-10-28"
---

Optics 104 · Cavity design

# Stable Resonator & Transverse Modes

**How should two mirrors be curved and spaced to make a stable cavity that favors the fundamental mode?** A two-mirror cavity is stable when the product of its two g parameters lies between zero and one; each g is one minus the cavity length over that mirror's radius of curvature. In that range a Gaussian beam stays confined over many round trips. An aperture of the right size then costs the wider higher-order modes more loss than the fundamental, so the fundamental mode takes over.

Space and curve two mirrors so a Gaussian mode fits, then size the iris so only the fundamental mode survives. Watch higher-order modes fade on a camera.

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

Two spherical mirrors face each other in air; concave radii count as positive and the beam is scalar and paraxial. With g equal to one minus the cavity length over a mirror radius, the cavity holds a Gaussian mode when the product of the two g values lies strictly between zero and one. Mode radii at the mirrors, the waist and its position follow Kogelnik and Li (1966); the radius is where the intensity falls to one over e squared of its peak. A product within one ten-thousandth of zero or one counts as the stability edge, where the mirrors no longer fix one mode size, and no mode is drawn there. Outside the stable region an off-axis ray is traced with the same mirror matrices until it leaves the mirrors. Tilting the output mirror about its vertex moves its centre of curvature; the new axis joins both centres, so it shifts by the cavity length times the tilt divided by one minus the product on the rear mirror, and by the rear g value times that at the iris. The mode keeps its shape about the shifted axis. The iris sits against the output mirror, so each round trip clips each mode once. The record follows the fifteen modes of order four or less, Laguerre–Gaussian petals for the round iris and Hermite–Gaussian modes for the square aperture, all starting with the same seed power. Each mode's loss is the power of the undistorted mode that falls outside the aperture, an estimate that leaves out the diffraction reshaping a Fox–Li calculation of the truncated cavity would give and treats every mode on its own. Below saturation all modes see the same gain, which cancels in the shares, so each round trip multiplies a mode's power by its own aperture transmission. Gain saturation, spatial hole burning, gain shape, thermal lensing, astigmatism and output coupling are not included. The camera images the output-mirror plane; modes of different order run at different frequencies, so their intensities add. The ideal fundamental beside it is the aligned, unclipped mode of the same mirrors and is not part of the record. Beam widths and axis shifts are drawn 32 times larger than lengths, tilt 1000 times larger, and the drawn beam radius is the second-moment radius of the recorded mix.

Physics tutorial

## Design a cavity whose camera shows one clean spot

**Background** Light bouncing between two curved mirrors either settles into a beam of fixed shape or wanders off the mirrors. The mirrors decide which, and an iris then decides how many beam shapes can keep going.

**Why it matters** The record starts with fifteen transverse modes at equal power and follows them for many round trips. The camera behind the output mirror shows the mix turning into one round spot, or staying a blur.

Start with the essentials

- **Focus question**: Which mirror curvature, spacing and iris leave only the fundamental mode, and why does a tilt matter more in some cavities?
- **One-sentence intuition**: The product of the two g values sets whether a Gaussian mode fits and how large it is on each mirror. Every higher-order mode is wider than the fundamental mode, so an iris slightly larger than the fundamental mode clips the others much harder. Each round trip repeats that small difference in loss, so the fundamental takes over. Near the flat and concentric ends, where the product approaches one, the mode grows on both mirrors and a tilt moves its axis in proportion to one over one minus the product.

### Core mathematical model

#### When a Gaussian mode fits

$$
0<g_1g_2<1,\qquad g_i=1-\frac{L}{R_i}
$$

L is the mirror spacing and R\_i the radius of curvature of mirror i, concave positive. Outside this range a ray next to the axis walks off the mirrors.

#### Mode radius on each mirror

$$
\begin{gathered}w_1^2=\frac{\lambda L}{\pi}\sqrt{\frac{g_2}{g_1(1-g_1g_2)}}\\ w_2^2=\frac{\lambda L}{\pi}\sqrt{\frac{g_1}{g_2(1-g_1g_2)}}\end{gathered}
$$

Lambda is the wavelength. Both radii grow without limit as the product approaches one, at the flat and the concentric ends of the stable region.

#### Waist inside the cavity

$$
w_0^2=\frac{\lambda L}{\pi}\sqrt{\frac{g_1g_2(1-g_1g_2)}{(g_1+g_2-2g_1g_2)^2}}
$$

Near the concentric end the waist shrinks to a tiny focus while the spots on the mirrors grow.

#### What a tilt does to the axis

$$
\Delta x_1=\frac{L\,\theta_2}{1-g_1g_2},\qquad \Delta x_2=g_1\,\Delta x_1
$$

Tilting the output mirror by theta\_2 moves its centre of curvature, and the axis joins both centres. Delta x\_1 is the shift on the rear mirror and Delta x\_2 at the iris.

#### Clipping loss of the two lowest modes

$$
\begin{gathered}\ell_{00}=e^{-2a^2/w_2^2}\\ \ell_{01}=\left(1+\frac{2a^2}{w_2^2}\right)e^{-2a^2/w_2^2}\end{gathered}
$$

For a round iris of radius a centred on the mode, this is the power of each undistorted mode outside the iris on one round trip.

#### Shares after k round trips

$$
s_i(k)=\frac{(1-\ell_i)^k}{\sum_j(1-\ell_j)^k}
$$

With equal seeds and the same gain for every mode, each round trip multiplies a mode by its own transmission. A loss difference of three percent shrinks a mode's share about twentyfold in one hundred round trips.

#### Where the transverse modes sit in frequency

$$
\frac{\nu}{c/2L}=q+1+\frac{N+1}{\pi}\arccos\left(\pm\sqrt{g_1g_2}\right)
$$

q counts the half wavelengths along the axis and N is the transverse order. The sign follows the sign of the g values. Modes of different order sit at different frequencies, so the camera adds their intensities.

### Common difficulties

#### Stable does not mean low loss

**Typical misconception** Once the stability product lies between zero and one, the cavity has low loss.

**Better mental model** Stability only says a Gaussian mode fits. Its loss depends on how the mirrors and the iris compare with the mode size, which grows near both ends of the stable region.

#### A smaller iris is not always better

**Typical misconception** Closing the iris further always gives a cleaner fundamental mode.

**Better mental model** Below about one spot radius the iris clips the fundamental mode too, so the laser needs much more gain to run. The best iris leaves the fundamental almost untouched and only clips the wider modes.

#### Flat mirrors are not the easy choice

**Typical misconception** Two flat or nearly flat mirrors make the simplest, most forgiving cavity.

**Better mental model** Near the flat end the mode is large and a tiny tilt shifts the axis by many times more than in a cavity with curved mirrors, so alignment becomes critical.

### Run the experiment

1. 01

   #### Watch the fundamental win

   Keep the default preset, Matched iris: the fundamental wins, and watch one replay.

   **What to observe:** The camera starts with a wide blur of fifteen modes. Within a few dozen round trips it narrows to one round spot, and the beam in the cavity slims down with it.

2. 02

   #### Open the iris

   Load Open iris: every mode survives.

   **What to observe:** No mode loses a measurable amount, so all fifteen keep equal shares and the camera never clears up.

3. 03

   #### Close the iris too far

   Load Tight iris clips the fundamental too.

   **What to observe:** The fundamental wins within a few round trips, but it now loses almost a fifth of its power each time, which the gain would have to replace.

4. 04

   #### Tilt a nearly flat cavity

   Load Nearly flat mirrors: a tilt walks the mode off, then drag the tilt slider back to zero.

   **What to observe:** Sixty microradians move the axis about half a spot radius, and the fundamental loss rises from below one percent to three percent. The default cavity barely notices the same tilt.

5. 05

   #### Leave the stable region

   From the default, drag the output mirror radius below the 30 cm cavity length.

   **What to observe:** The two g values now have opposite signs and the product turns negative. The beam disappears, and an off-axis ray walks off the mirrors after a few bounces.

Physical conclusion

Pick mirrors well inside the stable region, then make the iris about 1.6 times the spot radius: the fundamental loses under one percent per round trip while the next mode loses about four percent.

## Related experiments and articles

Continue exploring

### Confine and select a mode

A stable cavity supports light. What decides which spatial pattern grows?

1. [You are here **Stable Resonator & Transverse Modes** Curve and space two mirrors, then size an iris until only the fundamental mode reaches the camera.](https://sandboxphysics.com/labs/stable-resonator-transverse-modes/)
2. [Lab **Stable Resonator Geometry** Change both mirrors and inspect bounded rays and aperture loss.](https://sandboxphysics.com/labs/stable-resonator-geometry/)
3. [Lab **Transverse Laser Modes** Let pump overlap and saturation select competing modes.](https://sandboxphysics.com/labs/transverse-laser-modes/)

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 **Passive Optical Resonator** Compare stored light with self-sustained oscillation](https://sandboxphysics.com/labs/passive-optical-resonator/)
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)