Resonator Stability and the g-Parameter Diagram

The g-diagram is the first sketch on the whiteboard whenever someone designs a laser. A titanium-sapphire mode-locked oscillator, the workhorse of ultrafast science, is a folded cavity whose curved mirrors place a tight waist inside the crystal for strong Kerr-lensing while keeping the resonator just inside the stable region — and Kerr-lens mode-locking works by exploiting how the operating point shifts on the diagram between the continuous-wave and pulsed states, so a soft aperture favours the pulse. High-power solid-state lasers live at the opposite extreme: their gain rods develop a thermal lens that changes $R$, hence $g$, as pump power rises, and the design brief is to keep the operating point from wandering across a boundary over the full power range — a trajectory drawn directly on the g-plane. At the precision frontier the same criterion governs the kilometre-scale Fabry–Pérot arms of gravitational-wave detectors such as LIGO and Virgo, where mirror curvatures are engineered to hundreds of parts per million so the resonator sits at a safe point with a large, well-defined spot on each test mass. And where a large mode beats a clean Gaussian — high-energy pulsed CO$_2$ and solid-state systems — engineers step outside the stable region on purpose, using an unstable resonator whose magnifying round trip fills the whole gain aperture. In every case the first question is the one this article answers: where does the cavity sit in the $g_1$–$g_2$ plane, and how much margin does it have?

Treat one round trip as a single lens-like operation that keeps re-imaging the beam onto itself. A stable resonator is one in which a ray launched near the axis bounces back and forth forever, oscillating about the centre but never running away; an unstable one magnifies the ray's transverse excursion on every pass until it walks off the edge of a mirror. The g-parameters measure this focusing balance. Too little net focusing — mirrors too flat or too far apart — and the beam diffracts wider each pass and leaks out the sides. Too much — mirrors so strongly curved that the geometry drives past concentric — and the beam over-focuses and diverges. Between the two lies a corridor where the round-trip focusing is just restoring enough to hold a bounded, self-reproducing beam. A homely analogy is a marble rolling between two curved bowls: if the combined curvature nudges it back toward the centre each pass its path stays bounded, but if the geometry defocuses on net, any small nudge grows without limit. The product $g_1 g_2$ is the single parameter deciding between oscillatory (bounded) and hyperbolic (runaway) ray dynamics — and the same product decides whether a Gaussian mode has a real spot size or an imaginary one.

The self-consistency picture is complementary and actually fixes the mode. Insist that the Gaussian's parameter $q$ come back to itself after a full round trip: that is an eigenvalue problem for the round-trip matrix, and a confined mode with a real waist exists only when the eigenvalues are complex conjugates on the unit circle, $e^{\pm i\theta}$, rather than a real magnifying pair. That condition is $|A+D| \le 2$, nothing other than $0 \le g_1 g_2 \le 1$. Distance to the boundary carries real physics: near $g_1g_2 \to 1$ or $\to 0$ the spot swells to overfill the mirrors, diffraction loss spikes, and the slightest tilt throws the mode off. The chart below turns three lengths into one point and shows at a glance how much margin that point has.

The whole apparatus rests on one definition. Each mirror gets a dimensionless g-parameter measuring how strongly its curvature focuses relative to the cavity length: a flat mirror gives $g = 1$, one with radius equal to the spacing gives $g = 0$. The sign convention is fixed here: $R > 0$ for a mirror concave toward the cavity.

where: $g_i$ — dimensionless stability parameter of mirror $i$; $L$ — on-axis mirror separation; $R_i$ — radius of curvature of mirror $i$, positive if concave toward the cavity. Physically: as $L$ grows toward $R_i$ the focusing uses up the cavity length and $g_i \to 0$; push past $R_i$ and $g_i$ goes negative. Everything geometric about stability funnels into the product of the two g-parameters — the single discriminant. Not $g_1$ and $g_2$ separately, only their product decides whether a confined mode exists. It is the master result, the boundary of the diagram.

where: $g_1 g_2$ — product of the two g-parameters; a value in $[0,1]$ is necessary and sufficient for a confined Gaussian mode. At $g_1g_2 = 0$ or $1$ the cavity is marginally stable; values outside make it unstable. The bridge from the matrix to this inequality is the round-trip half-trace. Any lossless round-trip matrix is unimodular ($AD - BC = 1$), so its eigenvalues multiply to one: either a conjugate pair on the unit circle (stable) or a reciprocal real pair (unstable), and the half-trace decides which. Multiplying out the four matrices of a two-mirror trip gives the compact result below.

where: $A, D$ — diagonal elements of the round-trip matrix; $\theta$ — transverse round-trip (Gouy) phase per pass. Stability is $|\cos\theta| \le 1$, i.e. $-1 \le 2g_1g_2 - 1 \le 1$, exactly equation (2); the identity $\theta = 2\arccos\sqrt{g_1 g_2}$ ties the geometry to the transverse-mode spacing. To connect the matrix to an actual beam we need the object a round trip acts on. A Gaussian beam is fully described at any plane by one complex number, the beam parameter $q$, whose real and imaginary parts package the wavefront curvature and the spot size.

where: $q$ — complex beam parameter; $R$ — wavefront radius of curvature; $w$ — beam radius ($1/e^2$-intensity spot size); $\lambda$ — wavelength in the medium. The imaginary part makes the spot real and finite; a mode with $\mathrm{Im}(1/q) = 0$ is a plane wave, not a confined beam. A round trip transforms $q$ by the fractional-linear map governing any ABCD system. A self-reproducing mode is a fixed point of this map — the beam that comes back identical to itself.

where: $A, B, C, D$ — round-trip matrix elements; $q_1, q_2$ — beam parameters before and after the trip. Setting $q_2 = q_1$ gives a quadratic with complex roots — a physical, finite-waist mode — precisely when $|A+D| \le 2$, recovering equation (2). Solving that fixed-point condition gives the mode radii on the two mirrors — how the diagram earns its keep, turning a $(g_1, g_2)$ point into the actual spot a designer must fit inside a mirror and match an external beam to.

where: $w_1, w_2$ — mode radii at mirrors 1 and 2; $L, \lambda$ — spacing and wavelength. The factor $1/\sqrt{1-g_1g_2}$ is the tell-tale of the outer boundary: as $g_1g_2 \to 1$ both spots diverge and the mode overfills the mirrors. The internal beam waist rounds out the toolkit, collapsing to a memorable form for the symmetric confocal cavity where the Rayleigh range equals half the cavity length.

where: $w_0$ — intracavity waist radius; $w_{0,\text{conf}}$ — waist of the symmetric confocal resonator ($g_1 = g_2 = 0$, $R_1=R_2=L$); $L, \lambda$ — length and wavelength. The confocal waist is the smallest mode any symmetric stable cavity of length $L$ can hold — the natural mode-matching reference.

The named resonators are landmarks — $(g_1, g_2)$ points where the geometry has a familiar name and the mode behaviour is worth memorizing.

1 · Mode sizes of a hemispherical HeNe cavity (confidence builder)

A helium–neon laser at $\lambda = 632.8$ nm uses a flat output coupler ($R_1 = \infty$) and a concave back mirror $R_2 = 0.30$ m, separated by $L = 0.25$ m. Is it stable, and how big is the mode on each mirror?

Step 1. Compute the g-parameters. The flat mirror gives $g_1 = 1 - 0 = 1$; the curved mirror gives $g_2 = 1 - 0.25/0.30 = 0.1667$. The product $g_1 g_2 = 0.1667$ sits inside $[0,1]$, so the cavity is stable — it plots on the $g_1 = 1$ line just inside the boundary, the red HeNe point above.

Step 2. Because $g_1 = 1$, the wavefront is flat there, so the waist sits on the flat mirror and $w_0 = w_1$. With $L\lambda/\pi = (0.25)(632.8\times10^{-9})/\pi = 5.036\times10^{-8}\ \mathrm{m}^2$, equation (6) gives $\sqrt{0.1667/0.8333} = 0.4472$, so $w_1^2 = 5.036\times10^{-8}\times0.4472 = 2.252\times10^{-8}\ \mathrm{m}^2$ and $w_1 = w_0 = \mathbf{0.150\ mm}$.

Step 3. The spot on the curved mirror uses the companion formula: $\sqrt{g_1/[g_2(1-g_1g_2)]} = \sqrt{1/(0.1667\times0.8333)} = \sqrt{7.20} = 2.683$, so $w_2^2 = 5.036\times10^{-8}\times2.683 = 1.351\times10^{-7}\ \mathrm{m}^2$ and $w_2 = \mathbf{0.368\ mm}$. The mode fans from a $0.15$ mm waist on the flat mirror to a $0.37$ mm spot on the curved one. Push $L$ toward $0.30$ m and $g_2 \to 0$: the cavity nears the hemispherical boundary, the mode swells, and alignment gets touchy — the tuning knob every HeNe builder knows. Stable, with $w_0 = 0.150$ mm at the flat mirror and $0.368$ mm at the curved mirror.

2 · An asymmetric cavity that fails the sign test

A Nd:YAG design ($\lambda = 1064$ nm) proposes mirrors $R_1 = 1.0$ m and $R_2 = 0.5$ m at spacing $L = 0.8$ m. Will it lase on a clean mode? If not, fix it.

Step 1. Evaluate the g-parameters. $g_1 = 1 - 0.8/1.0 = 0.20$; $g_2 = 1 - 0.8/0.5 = -0.60$. The product $g_1 g_2 = -0.12$ is negative — outside $[0,1]$. The cavity is unstable: no confined Gaussian mode, so no clean TEM$_{00}$ beam. The point lands in a product-negative quadrant, where the green leaves never reach.

Step 2. Diagnose the culprit. Mirror 2 has $R_2 = 0.5$ m but $L = 0.8$ m $> R_2$, driving $g_2$ negative; one negative and one positive $g$ forces a negative product. Shorten the cavity so both g's share a sign. Try $L = 0.4$ m: $g_1 = 0.60$, $g_2 = 1 - 0.4/0.5 = 0.20$, product $g_1 g_2 = 0.12 \in [0,1]$ — now stable.

Step 3. Size the repaired mode at $\lambda = 1064$ nm, $L = 0.4$ m, with $L\lambda/\pi = 1.355\times10^{-7}\ \mathrm{m}^2$ and $1 - g_1g_2 = 0.88$. Mirror 1: $\sqrt{0.20/(0.60\times0.88)} = 0.6155$, so $w_1 = \mathbf{0.289\ mm}$. Mirror 2: $\sqrt{0.60/(0.20\times0.88)} = 1.846$, so $w_2 = \mathbf{0.500\ mm}$. A thermal lens in the pumped rod effectively shortens $R$ and shifts both g's, so a high-power design keeps margin here against drifting across the boundary as pump power ramps. Original cavity unstable ($g_1g_2 = -0.12$); shortening to $L = 0.4$ m restores stability ($g_1g_2 = 0.12$), giving mode radii $0.289$ mm and $0.500$ mm.

The stability diagram looks timeless, but it is barely older than the laser. After 1958 the open resonator — two mirrors with the sides thrown wide open — was a genuine puzzle: unlike a closed microwave cavity it has a continuum of diffraction loss, and it was not obvious that any low-loss mode should survive. The people who answered that built the entire language this article speaks.

Texts. Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 10–11 (Resonator Optics) — the cleanest modern treatment of the g-parameters · Siegman, Lasers, Ch. 19–21 (stable and unstable resonator theory) — the definitive reference · Hecht, Optics, 5th ed., §13.1 (laser resonator geometry) · Milonni and Eberly, Laser Physics, Ch. 7 (cavity modes) · Demtröder, Laser Spectroscopy, Vol. 1, §5.2 (open optical resonators) · Svelto, Principles of Lasers, Ch. 5 (passive optical resonators).

Provenance. The definition $g_i = 1 - L/R_i$ and the inequality $0 \le g_1 g_2 \le 1$ follow Kogelnik and Li's 1966 review and are reproduced in Saleh and Teich (Ch. 11) and Siegman (Ch. 19); the ABCD self-consistency route is standard in both. The named-cavity landmarks and the confocal waist $w_0 = \sqrt{L\lambda/2\pi}$ trace to Boyd and Gordon (1961) and Boyd and Kogelnik (1962). A verifier should confirm: (1) multiplying the four round-trip ray matrices symbolically gives half-trace $\tfrac12(A+D) = 2g_1g_2 - 1$ (checked here numerically for symmetric and asymmetric cases); (2) the Gaussian self-consistency discriminant $(A+D)^2 - 4 < 0$ yields the same inequality; (3) the confocal waist from $z_R = L/2$; (4) the worked mode sizes $w_1 = 0.150$ mm, $w_2 = 0.368$ mm for the HeNe cavity and the stability sign flip in the Nd:YAG example. No numerical constants beyond standard textbook geometry are asserted.

Prerequisites. Gaussian Beams · ABCD Matrix Beam Propagation · Optical Resonators and Cavity Modes. Leads to. Transverse Cavity Modes (Hermite–Gaussian) · Laser Threshold Condition · Cavity Q Factor and Photon Lifetime.

Ask the tutor: Give me your mirror radii and spacing and I will plot your cavity on the g-diagram, tell you how much stability margin it has, and size the mode on each mirror.