The two mirrors that hold a beam

The first laser, built by Theodore Maiman in 1960, was a rod of ruby with its two ends polished flat and parallel and coated with silver, one of them thinly enough to let some light out. The light inside bounced between the two flat ends, amplified on each pass, and a beam emerged from the thin end. The design was taken from the Fabry–Pérot interferometer, which had used two flat parallel mirrors since the 1890s, and it worked. It also turned out to be about the worst possible choice of mirrors for the purpose, for a reason that a single number exposes.

Within a year, Gardner Fox and Tingye Li at Bell Laboratories had computed, by brute numerical iteration, how a beam settles down between two flat mirrors over hundreds of round trips, and Gary Boyd and James Gordon had shown that two curved mirrors, spaced by their common radius of curvature, held a beam far better and were far easier to align. By 1966 Herwig Kogelnik and Li had organised every possible pair of mirrors on a single diagram. The diagram is the subject here, and it begins with a ray bouncing back and forth.

A cavity is a row of lenses #

A curved mirror reflects a ray as a lens of focal length $R/2$ would transmit it, with the direction reversed. So a ray bouncing between two mirrors of radii $R_1$ and $R_2$ a distance $L$ apart is equivalent, if the cavity is unfolded, to a ray passing along an endless row of lenses of focal lengths $R_1/2$ and $R_2/2$ alternating at spacing $L$. What a lens is doing found that a thin lens changes a ray’s slope in proportion to its height and leaves its height alone, and a gap changes its height in proportion to its slope and leaves its slope alone. Both are linear, so each can be written as a two-by-two matrix acting on the pair (height, slope), and a round trip of the cavity is the product of four matrices.

Whatever the matrix, a ray after $n$ round trips is the matrix raised to the $n$th power acting on the starting ray, and what a matrix’s powers do is decided by its eigenvalues. The round-trip matrix has determinant one — it conserves the area a bundle of rays occupies in the (height, slope) plane, the same invariant the invariant that is a count traced through every passive optical system — so its two eigenvalues multiply to one. Either they are a complex pair on the unit circle, $e^{\pm i\theta}$, and the ray oscillates for ever with bounded height; or they are real, $\lambda$ and $1/\lambda$ with $|\lambda| > 1$, and the ray grows geometrically and leaves. Which one happens is decided by the trace: the eigenvalues lie on the unit circle exactly when half the trace lies between −1 and 1.

One number for every pair of mirrors #

For two mirrors, the half-trace of the round trip comes out in a form that makes the whole problem visible. Define for each mirror

$$g_i = 1 - \frac{L}{R_i},$$

so that a flat mirror has $g = 1$ and a mirror whose centre of curvature lies at the other mirror has $g = 0$. Then the half-trace of the round trip is $2g_1g_2 - 1$, and the condition that it lie between −1 and 1 is

$$0 \le g_1 g_2 \le 1.$$

The diagram is the whole design space of two-mirror cavities, and its striking feature is where the familiar designs sit. Two flat mirrors are at (1, 1), exactly on the hyperbola. The confocal cavity, with the mirrors spaced by their radius so that their foci coincide, is at the origin, where the two stable regions touch at a corner. The concentric cavity, spaced by twice the radius so that the mirrors share a centre, is at (−1, −1), on the other hyperbola. The hemispherical cavity, a flat mirror at the centre of curvature of a curved one, is at (1, 0), on both an axis and a hyperbola at once. Every simple, symmetric, memorable design is marginal.

That is no coincidence. A marginal cavity is one whose round trip maps rays back onto themselves exactly, which is what makes it easy to describe — and exactly what makes it fragile. On the boundary, the ray’s oscillation has zero frequency, or a frequency that locks onto the round trip, and the slightest error in spacing or curvature tips it one way or the other. Real lasers sit a little way inside the shaded region.

Flat mirrors and their infinite beam #

The ray figure shows what nearly flat mirrors do. At $g = 0.99$ — mirrors of radius a hundred times their spacing — a ray is trapped, but it swings across the axis only once every 44 bounces and reaches further out than at $g = 0.5$. As $g$ approaches one the swing slows without limit, and at exactly one, flat mirrors, a ray that starts tilted simply walks sideways at the same rate on every bounce, never to return.

That is the ray picture. In the wave picture, the cavity holds a beam — a Gaussian mode, the narrowest pattern that reproduces itself after a round trip, with the profile and the spreading of the fan of plane waves inside every beam. Its size is set by the requirement that the beam’s wavefronts match the mirrors’ curvatures where it meets them.

For two identical mirrors the beam’s radius on the mirrors is

$$w^2 = \frac{\lambda L}{\pi}\,\frac{1}{\sqrt{1 - g^2}},$$

which is smallest for the confocal cavity, $g = 0$, and infinite at both edges of stability. Two flat mirrors would hold a beam of infinite width — or rather, since no mirror is infinite, they hold whatever the mirrors’ edges cut out of it, with diffraction losing light round the edges on every pass. That was the problem Fox and Li computed: flat-mirror cavities do settle into modes, but modes shaped by the mirrors’ edges and lossy in proportion to how small the mirrors are compared with the beam’s natural spread. Curved mirrors hold a beam much narrower than themselves, which leaks almost nothing past their edges.

At the other edge, the concentric cavity focuses its beam to a point at the centre while spreading it over the whole of each mirror. That is useful when a tiny, intense focus is wanted inside the cavity — and dangerous, because the beam’s waist is where the beam that focuses itself would begin to collapse if the medium there were glass.

The ray’s swing is the beam’s phase #

The angle $\theta$ by which a trapped ray advances around its oscillation on each round trip — from $\cos\theta = 2g_1g_2 - 1$ — has a second meaning that ties the ray picture to the wave picture exactly. The half cycle a focus adds found that a beam focused inside a cavity gains a Gouy phase on every pass, and that the cavity’s transverse modes resonate at frequencies shifted by that phase. The round-trip Gouy phase of the fundamental mode is precisely $\theta$. For the confocal cavity, $\theta = \pi$: a ray retraces itself after two round trips, and the transverse modes fall exactly halfway between the longitudinal ones. For nearly flat mirrors $\theta$ is small, the ray swings slowly, and the transverse modes crowd together in frequency.

So the slow swing of the ray at $g = 0.99$ and the near-degeneracy of a flat cavity’s transverse patterns are the same fact, seen once as geometry and once as interference. At the boundary of stability both degenerate together: the ray’s swing stops, and the transverse modes fall on top of one another, so that the cavity can no longer tell one pattern of light from another. That is a second reason flat-mirror lasers were unruly — their output wandered between patterns because nothing in the cavity preferred one.

The cavity that cannot be aligned #

The stability number also decides how much the beam cares about the mirrors’ alignment. Tilt one mirror slightly, and the cavity’s axis — the one ray that retraces itself — moves to a new line through the two centres of curvature.

The axis passes through the two mirrors’ centres of curvature, and how far it moves when a mirror tilts depends on how far apart those centres are. For a confocal cavity they are a cavity length apart and the axis pivots stiffly; as the mirrors flatten, the centres recede to infinity on opposite sides, the line through them becomes ill-defined, and the smallest tilt swings it wildly. The shift goes as $1/(1 - g^2)$, so at $g = 0.99$ a tilt of ten microradians — the angle a hair subtends at a metre — moves the beam on the far mirror by 151 micrometres, a substantial fraction of a beam a millimetre across. At $g = 1$ the shift is infinite: two flat mirrors cannot be aligned at all, only made parallel to within the beam’s own diffraction angle, which is why Maiman’s ruby had to be polished flat and parallel to a few seconds of arc.

The same trace, four other places #

The condition that half the trace of a period’s matrix lie between −1 and 1 is not a fact about mirrors. It is the general condition for stability of any linear system repeated periodically, Floquet’s theorem in its simplest form, and it has turned up in this collection several times under different names.

The swing that is pumped, not pushed is a pendulum whose length is varied periodically; over each period its (angle, rate) is multiplied by a matrix, and when the trace leaves the band the swing grows exponentially — the parametric resonance tongues. Held up by a force that averages to nothing, the inverted pendulum stabilised by shaking its pivot, is the same calculation with the trace brought back inside the band. The gap a repeat opens is a wave crossing a crystal, cell by cell: inside the band the trace condition holds and the wave propagates; outside it, the wave decays exponentially and the crystal reflects it — the band gap is the unstable region of the stability diagram.

And in 1952 Ernest Courant, Stanley Livingston and Hartmut Snyder realised that a particle accelerator could focus its beam far more strongly by alternating magnets that focus in one plane and defocus in the other, provided the alternation kept the trace of each cell’s matrix inside the band — strong focusing, which made every large accelerator since possible. A laser cavity, an accelerator lattice, a crystal and a pumped swing are one calculation: a two-by-two matrix of determinant one, raised to a large power.

A pipe of lenses across New Jersey #

Before optical fibres, the same diagram was the design rule for long-distance optical communication. In the early 1960s, engineers at Bell Laboratories proposed carrying light in buried pipes, refocused by a lens every hundred metres or so — a lens waveguide, exactly the unfolded cavity of the first figure, with the stability condition becoming the requirement that the lens spacing be no more than four focal lengths. Goubau and Schwering had worked out the beam such a line would carry, and test lines were built at the laboratories’ Holmdel site.

They failed for the reasons the tilt figure predicts. A line of hundreds of lenses is a cavity that never closes, and each lens’s misalignment is passed on to the next and accumulates; the ground settled, the pipes bent by fractions of a millimetre, and the beam wandered off the axis. Worse, the air inside the pipe developed temperature gradients whenever the sun warmed one side of it, and the gradients bent the light like the ray that bends without a surface — a mirage inside the pipe. The engineers turned the defect into a device, building deliberate gas lenses out of heated tubes of flowing gas. The problem dissolved in 1970, when Corning made a glass fibre with losses low enough to replace the whole apparatus, and the fibre’s graded core does continuously what the row of lenses did in steps.

A cavity built to be unstable #

The diagram’s unshaded region is not useless. In 1965 Anthony Siegman pointed out that a deliberately unstable cavity has properties a high-power laser wants. In a stable cavity the beam is narrow — a fraction of a millimetre for a metre-long cavity — and it can only extract energy from the thin column of gain medium it fills. In an unstable cavity a ray is magnified on every round trip and walks outward until it escapes past the edge of the smaller mirror, so the light fills the entire medium, however wide.

The output is an annulus of light around the small mirror, and the fraction let out per round trip is set by the magnification — 56 per cent at a magnification of 1.5 for a round mirror. That is far too lossy for a low-gain laser such as a helium–neon tube, which gains a few per cent per pass and needs a stable cavity that keeps the light for hundreds of trips. It is ideal for carbon-dioxide, chemical and excimer lasers with large, high-gain media, which amplify enough in a pass to make up a loss of half. A ray in the unstable cavity grows geometrically, the same exponential the slightly convex mirrors produced in the ray figure, and the cavity turns that instability into a controlled way of getting the light out.

Paraxial rays between perfect mirrors, with no gain inside #

Every figure is paraxial: rays close to the axis and nearly parallel to it, mirrors described by their curvature alone. Steep rays see a spherical mirror’s aberration, which the mirror that cannot focus found, and the stability boundary is blurred for them. The beam-size figure assumes mirrors much larger than the beam, so that their edges play no part; near the boundaries, where the beam grows towards the mirror’s size, diffraction at the edges takes over, and what the cavity holds is set by its Fresnel number — the mirror’s area divided by the wavelength times the spacing — rather than by $g$.

Nothing in the figures includes the gain medium, which in a real laser is itself a lens. A pumped rod heats unevenly and develops a thermal lens that changes with the pump power, moving the cavity’s working point across the diagram as the laser warms up — a cavity designed at the centre of the stable region at low power can be driven out of it at high power. Mirrors tilted to fold the beam behave as cylindrical lenses of different strengths in the two planes, so a folded cavity has two stability numbers, one per plane. And the tilt figure treats a static misalignment; vibrations at the cavity’s own frequencies are a different and harder problem.

Still open: the mirrors that the light itself moves #

In the most sensitive cavities ever built, the light pushes the mirrors. The arms of the gravitational-wave detectors are optical cavities four kilometres long, storing hundreds of kilowatts of light on mirrors of forty kilograms hung as pendulums. The radiation pressure of that light exerts a torque on a mirror whenever the beam is off centre, and since tilting a mirror moves the beam, as the tilt figure shows, the light and the tilt form a feedback loop. Depending on the cavity’s $g$-factors, one angular motion is stiffened and another is softened, and above a certain power the softened one becomes unstable — the Sidles–Sigg instability, which the detectors’ control systems must suppress.

Similar loops couple the stored light to the mirrors’ internal acoustic modes, converting optical power into ringing at tens of kilohertz. How much power such cavities can store before these optomechanical instabilities become uncontrollable, and how to design $g$-factors and mirror shapes to push the limit back, are active questions for the next generation of detectors.

The two mirrors of a laser are the simplest place to see the general rule. A system repeated periodically — a cavity, a crystal, a lattice of magnets, a pumped swing — is stable exactly when half the trace of one period’s matrix lies between −1 and 1; and the simplest, most symmetric designs sit on the edge of that band. Two flat mirrors look like the obvious way to send light back and forth. They are the one arrangement that cannot hold it.