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.

Stability product of the two mirrors $\text{—}$

Fundamental mode: loss per round trip at the iris $\text{—}$

Lowest-loss higher-order mode $\text{—}$

Fundamental mode share at the end of the record $\text{—}$

Mode radius on the rear / output mirror $\text{—}$

Waist radius and its distance from the rear mirror $\text{—}$

Axis shift on the rear mirror / at the iris $\text{—}$

Gouy phase per round trip $\text{—}$

Transverse mode spacing / free spectral range $\text{—}$

Fresnel number of the iris $\text{—}$

Round trips until the fundamental holds half the power $\text{—}$

Mean transverse order at the end $\text{—}$

Round trips recorded $\text{—}$

Optics 104

How should two mirrors be curved and spaced to make a stable cavity that favors the fundamental mode?

Cavity length

L

30\,\mathrm{cm}

Rear mirror radius of curvature

R_1

100\,\mathrm{cm}

Output mirror radius of curvature

R_2

100\,\mathrm{cm}

Iris radius at the output mirror

a

0.6\,\mathrm{mm}

Output mirror tilt

\theta_2

0\,\mathrm{\mu rad}

Vacuum wavelength

\lambda

1064\,\mathrm{nm}

Aperture shape

S

0

Round trips recorded

N

200

Every change records a new competition from equal seeds in all fifteen modes. The record is computed once; the replay never recomputes it.

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.

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 misconceptionOnce the stability product lies between zero and one, the cavity has low loss.

Better mental modelStability 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 misconceptionClosing the iris further always gives a cleaner fundamental mode.

Better mental modelBelow 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 misconceptionTwo flat or nearly flat mirrors make the simplest, most forgiving cavity.

Better mental modelNear 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

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.
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.
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.
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.
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.

Stable Resonator & Transverse Modes

Design a cavity whose camera shows one clean spot

Core mathematical model

When a Gaussian mode fits

Mode radius on each mirror

Waist inside the cavity

What a tilt does to the axis

Clipping loss of the two lowest modes

Shares after k round trips

Where the transverse modes sit in frequency

Common difficulties

Stable does not mean low loss

A smaller iris is not always better

Flat mirrors are not the easy choice

Run the experiment

Watch the fundamental win

Open the iris

Close the iris too far

Tilt a nearly flat cavity

Leave the stable region

Related experiments and articles

Confine and select a mode

Controlled light and precision measurement