Resonator stability

Resonator stability is the property of an optical cavity that decides whether light bouncing between its mirrors is refocused toward the axis or walks off after a few passes. In a stable resonator a paraxial ray stays within a bounded distance of the axis indefinitely, and the cavity supports a set of Gaussian modes whose size and wavefront curvature reproduce themselves after each round trip. In an unstable resonator rays diverge geometrically and leave past the mirror edges as loss, or as deliberate output coupling in some lasers. Most low- and medium-gain lasers use stable resonators.

The g-parameter condition

For a round-trip ray matrix with elements $A$ and $D$, rays stay bounded when $-1 < (A+D)/2 < 1$; the ABCD matrix entry derives this. For two mirrors of radii $R_1$ and $R_2$ (positive for concave) a distance $L$ apart, it reduces to

$$0 < g_1 g_2 < 1,\qquad g_i = 1 - \frac{L}{R_i}.$$

Plotted in the $(g_1, g_2)$ plane, the stable region lies between the axes and the hyperbola $g_1g_2 = 1$. The plane-parallel ($g_1 = g_2 = 1$), symmetric confocal ($g_1 = g_2 = 0$) and symmetric concentric ($g_1 = g_2 = -1$) resonators all sit on the boundary, where a small change in mirror curvature, or in length for the concentric case, can make the cavity unstable.

Mode size and a worked example

The fundamental Gaussian beam mode of a stable two-mirror cavity has radii on the mirrors

$$w_1^2 = \frac{L\lambda}{\pi} \sqrt{\frac{g_2}{g_1\,(1 - g_1 g_2)}},$$

with $w_2$ obtained by exchanging the indices. Take a plano-concave cavity of length 0.3 m with a flat mirror ($g_1 = 1$) and a concave mirror of radius 0.5 m ($g_2 = 0.4$), at 1064 nm. Then $g_1g_2 = 0.4$, the mode radius is 288 µm at the flat mirror, which is also the waist, and 455 µm at the curved mirror. Lengthening the cavity to 0.49 m brings it close to the hemispherical limit $L = R$: $g_1g_2 = 0.02$, the waist shrinks to 154 µm and the spot on the curved mirror grows to 1.09 mm. Near any boundary of the stability diagram the mode size becomes very sensitive to length and curvature, and the axis becomes sensitive to mirror tilt.

The same parameters fix the frequencies of the higher-order transverse modes through the Gouy phase. The spacing between adjacent transverse orders is

$$\Delta\nu_T = \frac{c}{2L}\,\frac{\arccos\sqrt{g_1 g_2}}{\pi}.$$

For the 0.3 m example the longitudinal spacing is 500 MHz and $\Delta\nu_T$ is 140.9 MHz. Resonators with rational values of this ratio, such as the confocal cavity, have degenerate transverse modes; in a Fabry-Perot analyser this degeneracy is used on purpose.

Thermal lenses and design practice

Intracavity lenses change the effective g-parameters. A flat-flat cavity of length $L$ with a thin lens of focal length $f$ at its centre has $g_1 = g_2 = 1 - L/(2f)$, so it is stable for any $f > L/4$ except $f = L/2$, where $g = 0$ puts it on the boundary; for $L$ = 0.3 m that means $f > 75$ mm. Such a cavity is marginal with no lens and is stabilized by the positive thermal lens of a pumped rod. Because the thermal lens varies with pump power, solid-state laser resonators are designed to remain stable, and to keep the TEM₀₀ mode size matched to the pumped region, over the working range of lens power. Magni showed that the width of this stable range, expressed in dioptric power of the rod, scales inversely with the mode area in the rod, so a larger mode, desirable for power extraction, narrows it.

Measurement

Stability is checked by translating a mirror while monitoring output: the laser stops or its mode changes shape abruptly at the stability edges. Mode radii are measured with a beam profiler at known distances and compared with the predicted Gaussian. The transverse mode spacing appears as beat notes in the RF spectrum of a fast photodiode signal, giving a direct measure of $g_1g_2$.

Unstable resonators

For high-gain media with large apertures, such as carbon-dioxide, chemical and excimer lasers, a stable resonator would either support many transverse modes or confine a mode too small to fill the medium. Siegman's unstable resonator instead magnifies the beam by a factor $M$ per round trip and couples out the part that passes the smaller mirror. For spherical mirrors, magnifying in both transverse directions, the geometric output fraction is $1 - 1/M^2$, 75 % for $M = 2$. The near field is annular.

Common questions

What does g₁g₂ = 1 mean for a laser?

The resonator is on the stability boundary. For flat mirrors it is the plane-parallel cavity, whose modes are set by apertures, tilt and any thermal lens; small perturbations determine whether it behaves as stable or unstable.

Is the confocal resonator stable?

The symmetric confocal cavity, $L = R$, sits at $g_1 = g_2 = 0$, the centre of the diagram. Its mode size is least sensitive to small mirror changes, although it lies formally on the boundary lines $g_1 = 0$ and $g_2 = 0$, and its transverse modes are degenerate.

References: H. Kogelnik, T. Li, Laser beams and resonators, Appl. Opt. 5, 1550 (1966); A. E. Siegman, Unstable optical resonators for laser applications, Proc. IEEE 53, 277 (1965); A. E. Siegman, Lasers (University Science Books, 1986); V. Magni, Resonators for solid-state lasers with large-volume fundamental mode and high alignment stability, Appl. Opt. 25, 107 (1986).