L10 · Modes and cavity design

Stable Resonator Geometry

Change two mirror curvatures and their separation. Connect the stability map to an ABCD ray trace, Gaussian waist and two-pass aperture loss. Explore 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 exact confocal with a one-centimetre change to only one mirror. Trace a tilted ray in the plane-parallel cavity. Move or shrink the aperture without changing geometric stability.

Aligned spherical mirrors in air, scalar paraxial optics, positive concave radii. The rear mirror defines axial zero; positive transverse height is upward, and ray slope follows the local propagation direction. Gaussian radius is the intensity one-over-e-squared radius. Stable interior points have a unique positive-imaginary fixed point of the round-trip matrix. Exact confocal geometry has a degenerate family; the displayed centered symmetric Gaussian is a selected member, not a unique solution. Other marginal points have no finite mode assigned. Aperture loss integrates the undeformed Gaussian twice per round trip, excluding diffraction reshaping, finite mirror apertures, gain guiding, thermal lenses, tilt and astigmatism. Rays stop at the physical aperture or when the paraxial limit is exceeded. Mirror sag and transverse dimensions are schematic; ray and Gaussian envelope share the same transverse magnification.

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: Interior eigenvalues have unit magnitude and bounded ray motion. At repeated eigenvalues, the full matrix matters: a confocal minus-identity map repeats rays, while a Jordan map accumulates displacement.

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 contains transverse height and local forward slope. A plane mirror has zero focusing power. The round trip starts immediately after the rear reflection.

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}

Strict interior stability requires the product between zero and one. Boundary products need separate matrix analysis rather than a blanket stable/unstable label.

Gaussian 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}

Select the solution with positive imaginary part. Its real part sets the waist location; its imaginary part sets the Rayleigh range. The radius and all scene envelopes come from this same solution.

Symmetric cavity limit

\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 independent limit checks the matrix solution. At the confocal point, the selected symmetric member has 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}

Every ray returns after two round trips in the ideal paraxial map. The fixed-point equation alone does not choose a unique beam; the symmetric displayed member is an explicit selection.

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 circular aperture is crossed twice. This is an integral of the unperturbed Gaussian intensity; severe truncation changes the mode and requires diffraction propagation.

Common difficulties

Unit eigenvalues are not enough on a boundary

Typical misconceptionEigenvalue magnitude one guarantees robust confinement.

Better mental modelA non-diagonalizable boundary map can grow polynomially. Exact confocal is diagonalizable but degenerate and sensitive to asymmetric perturbations.

Ray and Gaussian are different tests

Typical misconceptionA surviving ray proves the whole optical mode fits.

Better mental modelA single ray can pass an aperture that clips a substantial fraction of a Gaussian. The integrated loss and 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 undeformed reference. When loss is large, diffraction reshaping invalidates that assumption.

Run the experiment

01
Connect map and trajectory
Trace 32 passes in Stable symmetric. Change only the launch height or angle, then trace again.
What to observe: The Gaussian solution remains unchanged while the test ray follows a different bounded orbit, unless the aperture stops it.
02
Probe exact confocal
Choose Exact confocal and step four times. Change only one curvature by one centimetre in either direction.
What to observe: The exact ray repeats. Changing only one curvature gives a marginal Jordan map with accumulating deviations. Further changes to both curvatures can enter an unstable wedge.
03
Inspect boundaries
Compare Plane-parallel, Hemispherical and Concentric boundaries. Trace the same nonzero launch angle.
What to observe: Repeated eigenvalues can have magnitude one while deviations accumulate. No finite Gaussian is assigned to these non-confocal marginal cases.
04
Separate stability and clipping
Compare Near concentric and Stable but clipped. Move the aperture along the cavity, then increase wavelength.
What to observe: The Gaussian spot can be large near a boundary. Changing aperture or wavelength changes clipping while leaving the geometric stability product unchanged.