L10 · Modes and cavity design

Stable Resonator Geometry

Does a stable cavity guarantee that its beam fits through the aperture? No; stability and fit are two separate checks. A displaced ray stays near the axis if the mirror curvatures and spacing put the cavity inside its stability range. Even then, the Gaussian mode can be wider than the aperture, so check the mode size there too. Cases right on the edge of stability, such as the confocal cavity, need extra care.

Change the curvature of two mirrors and the distance between them. See how the stability map connects to an ABCD ray trace, the Gaussian waist and the aperture loss over two passes, and explore the confocal, plane-parallel, hemispherical and concentric boundaries.

Traced one-way passes $\text{—}$

Rear mirror stability parameter $\text{—}$

Output mirror stability parameter $\text{—}$

Stability product $\text{—}$

Geometry classification $\text{—}$

Round-trip matrix trace $\text{—}$

Largest eigenvalue magnitude $\text{—}$

Gaussian waist radius $\text{—}$

Waist distance from rear mirror $\text{—}$

Rayleigh range $\text{—}$

Rear mirror mode radius $\text{—}$

Output mirror mode radius $\text{—}$

Estimated round-trip clipping loss $\text{—}$

Traced ray status $\text{—}$

L10 · Geometry workbench

Follow the same ray through the cavity

Use a plane rear mirrorIgnore its curvature control and set focusing power to zero.Use a plane output mirrorIgnore its curvature control and set focusing power to zero.Mirror separation

L

50\,\mathrm{cm}

Rear mirror curvature radius

\mathcal R_1

100\,\mathrm{cm}

Output mirror curvature radius

\mathcal R_2

100\,\mathrm{cm}

Vacuum wavelength

\lambda

1064\,\mathrm{nm}

Aperture radius

a

0.6\,\mathrm{mm}

Aperture position along cavity

z_a/L

50\,\mathrm{\%}

Initial ray height at rear mirror

y_0

0.15\,\mathrm{mm}

Initial forward slope

\theta_0

0.4\,\mathrm{mrad}

Compare the exact confocal cavity with the case where only one mirror changes by one centimetre. Trace a tilted ray in the plane-parallel cavity. Then move or shrink the aperture: the clipping changes, but the geometric stability does not.

The model uses aligned spherical mirrors in air and scalar paraxial optics, with concave radii counted as positive. Axial position is measured from the rear mirror. Transverse height is positive upward, and the ray slope is taken along the local direction of travel. The Gaussian radius is where the intensity falls to one over e squared of its peak. Inside the stable region, the round-trip matrix has exactly one fixed point with a positive imaginary part. The exact confocal geometry has a whole family of solutions; the displayed centered symmetric Gaussian is one chosen member, not the only solution. No finite mode is assigned at the other boundary points. Aperture loss integrates the undistorted Gaussian twice per round trip. It leaves out diffraction reshaping, finite mirror apertures, gain guiding, thermal lensing, tilt and astigmatism. A ray stops when it hits the physical aperture or leaves the paraxial range. Mirror sag and transverse sizes are schematic, but the ray and the Gaussian outline share the same transverse magnification.

Physics tutorial

A cavity must reproduce its transverse field

BackgroundResonance along the axis fixes the frequencies, and the mirror curvatures decide whether the beam stays confined sideways. A Gaussian mode of the cavity comes back with the same complex beam parameter after every round trip.

Why it mattersThe stability product by itself hides what happens at the edges of the stable region. Following a ray, finding the Gaussian mode and estimating aperture loss each show something the others miss.

Start with the essentials

Focus question: How can a cavity be geometrically stable but strongly clipped, or exactly periodic without a unique Gaussian mode?
One-sentence intuition: Inside the stable region, the eigenvalues have magnitude one and rays stay bounded. Where the eigenvalues coincide, you need the whole matrix: the confocal map is minus the identity and brings every ray back, while a Jordan-form map lets the displacement keep growing.

Core mathematical model

Propagation and reflection

\begin{aligned}P(L)&=\begin{pmatrix}1&L\\0&1\end{pmatrix}\\M_i&=\begin{pmatrix}1&0\\-2/\mathcal R_i&1\end{pmatrix}\end{aligned}

The ray vector holds the height above the axis and the local forward slope. A plane mirror has no focusing power. Each round trip starts just after the reflection at the rear mirror.

Round-trip stability

\begin{aligned}M&=M_1P M_2P\\g_i&=1-L/\mathcal R_i\\\operatorname{tr}M&=4g_1g_2-2\\\det M&=1\end{aligned}

A cavity is strictly inside the stable region when the product lies between zero and one. At the boundary values, the matrix itself has to be examined; a simple stable or unstable label does not apply.

Gaussian mode as a fixed point

\begin{aligned}q&=\frac{Aq+B}{Cq+D}\\q(z)&=z-z_0+iz_R\\w^2(z)&=w_0^2\left[1+\left(\frac{z-z_0}{z_R}\right)^2\right]\\w_0^2&=\frac{\lambda z_R}{\pi}\end{aligned}

Take the solution with a positive imaginary part. Its real part gives the waist position, and its imaginary part gives the Rayleigh range. The beam radius and every beam outline in the scene come from this one solution.

Symmetric cavity check

\begin{aligned}z_0&=L/2\\z_R&=\frac12\sqrt{L(2\mathcal R-L)}\\w_0^2&=\frac{\lambda z_R}{\pi}\end{aligned}

This closed-form result is an independent check on the matrix solution. At the confocal point, the chosen symmetric mode has a Rayleigh range equal to half the cavity length.

Confocal degeneracy

\begin{aligned}\mathcal R_1&=\mathcal R_2=L\\M&=-\mathbb I\\M^2&=\mathbb I\end{aligned}

In the ideal paraxial map, every ray returns after two round trips. The fixed-point equation does not pick out a single beam, so the Lab explicitly chooses the symmetric one to display.

Two-pass aperture estimate

\begin{aligned}u_a&=\frac{2a^2}{w^2(z_a)}\\Q_a&=(1-e^{-u_a})^2\\\ell_a&=1-Q_a\end{aligned}

The beam crosses the circular aperture twice per round trip. The estimate integrates the undisturbed Gaussian intensity; strong clipping changes the mode itself and needs a diffraction calculation.

Common difficulties

Unit eigenvalues are not enough on a boundary

Typical misconceptionIf the eigenvalues have magnitude one, the beam is safely confined.

Better mental modelOn the boundary, a map that cannot be diagonalized can make displacements grow polynomially. The exact confocal map can be diagonalized, but it is degenerate and sensitive to asymmetric changes.

Ray and Gaussian are different tests

Typical misconceptionA surviving ray proves the whole optical mode fits.

Better mental modelOne ray can pass an aperture that cuts off a substantial part of the Gaussian beam. The integrated loss and the ray status answer different questions.

Aperture loss is not a field solution

Typical misconceptionThe clipped cavity retains exactly the same Gaussian eigenmode.

Better mental modelThe readout integrates an undistorted Gaussian reference. When the loss is large, diffraction reshapes the mode and that assumption fails.

Run the experiment

01
Link the matrix to the ray path
Trace 32 passes in Stable symmetric. Change only the launch height or angle, then trace again.
What to observe: The Gaussian mode stays the same. The test ray follows a different bounded path, unless the aperture stops it.
02
Test the exact confocal cavity
Choose Exact confocal and step four times. Then change the curvature radius of just one mirror by one centimetre, up or down.
What to observe: In the exact case the ray repeats. Changing just one curvature puts the cavity on the boundary with a Jordan-form map, and the deviations build up. Changing both curvatures further can take the cavity into an unstable wedge.
03
Look at the boundary cavities
Compare the Plane-parallel, Hemispherical and Concentric boundary presets, tracing each with the same nonzero launch angle.
What to observe: Repeated eigenvalues can have magnitude one even while the deviations build up. These boundary cases are not confocal, and no finite Gaussian mode is assigned to them.
04
Separate stability and clipping
Compare Near concentric and Stable but clipped. Move the aperture along the cavity, then increase wavelength.
What to observe: Near a boundary the Gaussian spot can be large. Changing the aperture or the wavelength changes the clipping but leaves the geometric stability product unchanged.