Astrophysics

The mirror held on a spring of light

Light bouncing between two mirrors pushes them apart, and how hard it pushes depends on how long the gap is, because the gap decides how much light the cavity holds. A mirror free to move therefore sits on a spring made of light: with a kilowatt stored in a metre-long cavity it is four-fifths as stiff as a steel wire a millimetre thick. The spring has a flaw no steel spring has. Stored light takes time to rise and fall, so the push always arrives a little late, and a late spring either winds the mirror up or calms it, depending only on which side of the cavity's resonance the laser sits.

17 min read 5 figures The same equation again Where the model stops

Assumes: Light has a pressure · The friction made of light

Light has a pressure put a number on the push of light against a mirror: twice the power divided by the speed of light, which for a watt is seven billionths of a newton. That number is so small that radiation pressure is usually filed under curiosities, the force that turns a solar sail over months. It stops being small when the light is used more than once. Put the mirror at the end of an optical cavity, facing a second mirror, and a photon bounces between them thousands of times before it leaks out, pushing on every bounce. A cavity with mirrors that lose one part in ten thousand per bounce holds, at a steady state, about three thousand times the power fed into it. Three-tenths of a watt from a laser becomes a kilowatt circulating, and a kilowatt pushes with nearly seven micronewtons.

That is still a small force. What makes it interesting is not its size but the fact that it depends on where the mirror is.

The power a cavity holds depends on where its mirror is. The power circulating in a cavity of finesse 10,000, a metre long and lit at 1064 nm, as a fraction of its value on resonance, against the detuning of the laser from the cavity's resonance in units of the cavity's half-linewidth (7.5 kHz). Lengthening the cavity lowers its resonance and moves every point of the curve to the right: 38 half-linewidths for each nanometre of mirror travel. On the blue side (laser above resonance, right) that motion loses power, so the radiation force on the mirror falls as the mirror moves out — a restoring spring, steepest at δ/γ = 0.58 (dashed tangent). On the red side the same motion gains power and pushes harder: a spring of the opposite sign. — Fig. 1 Power circulating in a metre-long cavity of finesse 10,000 lit at 1064 nm, as a fraction of its peak, against the detuning of the laser above the cavity’s resonance in half-linewidths (one half-linewidth is 7.5 kHz). A nanometre of mirror travel shifts the whole curve by 38 half-linewidths. On the blue side (right) moving the mirror out loses power and the push falls, a restoring spring steepest at $\delta/\gamma = 0.58$ ; on the red side the same motion raises the push.

Why the push depends on the gap

A cavity stores light only near its resonances, the frequencies at which a whole number of half-wavelengths fits between the mirrors. Two walls that let more through than one found that rule for the light passing through a pair of partial mirrors; the light trapped inside obeys the same one. Off resonance, successive round trips return out of step and cancel; on resonance they add. The stored power plotted against the laser’s frequency is the Lorentzian curve of the first figure, and its width is set by how fast light leaks out — the rule the width that is a lifetime found for every resonance, with the cavity’s storage time in place of an atom’s. For the cavity drawn here, a metre long with a finesse of ten thousand, the full width is 15 kilohertz on a light frequency of 280 terahertz. The curve is extraordinarily narrow.

Now move the mirror. A longer cavity fits its half-wavelengths at a slightly lower frequency, so its resonance slides down and the laser, whose frequency is fixed, finds itself further above it. The shift is large in units of the curve’s own width: a nanometre of travel moves the resonance by 38 half-linewidths, far across the whole peak. So the power the cavity holds, and with it the force on the mirror, changes steeply with the mirror’s position. Where a force changes with position there is a spring, and its stiffness is the slope of the force.

The sign of the slope depends on which side of the peak the laser sits. If the laser is tuned a little above the cavity’s resonance, on the blue side, lengthening the cavity moves it further from resonance: the stored power falls and the push on the mirror weakens. A mirror that has moved out is pushed out less, and one that has moved in, towards resonance, is pushed out more. That is a restoring force, a spring. On the red side of the peak everything reverses: moving out brings the cavity towards resonance, the push grows, and the mirror is driven further along whichever way it moved. That is an anti-spring, a force that pushes the mirror away from its equilibrium.

As stiff as a steel wire

The stiffness of a spring made of light. The spring constant of the radiation force on the cavity's end mirror, −dF/dx, against detuning, for 1 kW, 500 W, 100 W circulating. It is positive (restoring) on the blue side, negative on the red, zero on resonance, and largest at δ/γ = ±0.58: 1.63 × 10⁵ N/m at 1 kW, which is 81 per cent of the stiffness of a steel wire a metre long and a square millimetre thick (dashed). It scales with the power and with the square of the finesse. Held by this spring alone, a 1-gram mirror would oscillate at 2031 Hz — where the wire that hangs it, a pendulum, gives about 1 Hz. — Fig. 2 The spring constant of the radiation force on the end mirror against detuning, for 1 kW, 500 W and 100 W circulating. Positive on the blue side, negative on the red, zero on resonance, and largest at $\delta/\gamma = \pm 0.58$ , where it is $1.63 \times 10^5$ N/m at 1 kW, 81 per cent of a steel wire a metre long and a square millimetre in section (dashed). A 1-gram mirror held by it alone would oscillate at 2031 Hz.

The second figure turns the slopes into spring constants. The stiffness is greatest where the curve is steepest, at a detuning of $1/\sqrt3$ of the half-linewidth, and it scales with the power stored and with the sharpness of the resonance. At a kilowatt it reaches $1.6 \times 10^5$ newtons per metre. A steel wire a metre long and a square millimetre in section, pulled along its length, resists with $2 \times 10^5$ . The light in the cavity, weighing nothing and touching the mirror only by reflection, is four-fifths as stiff as the wire.

The comparison is fair in a second sense. The mirror is usually hung from fibres as a pendulum, so that it moves freely along the cavity’s axis, and that pendulum is soft: a swing period of a second or so, a spring constant for a 1-gram mirror of about four hundredths of a newton per metre. The optical spring is four million times stiffer. A 1-gram mirror held by it would oscillate at about two thousand hertz, and the pendulum it hangs from would hardly matter.

The spring needs no contact, adds none of the thermal noise that every material spring carries, and can be switched off, retuned or reversed by moving the laser’s frequency. Vladimir Braginsky, who worked out the mechanics of light in cavities in Moscow from the late 1960s, recognised it as a way to stiffen a test mass without touching it, and the idea has since become part of the design vocabulary of the gravitational-wave detectors, where detuning one of the interferometer’s cavities puts an optical spring on the mirrors and reshapes the band in which the instrument is most sensitive.

The push that arrives late

The figures so far treat the cavity as though it responded at once, and it does not. Light leaks out of the cavity at a definite rate — for this one the stored field decays by a factor of e in about 21 microseconds — and when the mirror moves, the stored power has to rise or fall to its new value at that rate. The force follows the mirror with a delay.

The push arrives after the mirror has moved. One cycle of the end mirror driven back and forth at 2 kHz through 100 × 10⁻¹⁵ m (solid), with the restoring part of the radiation force it produces, computed by integrating the cavity's field (dashed), both scaled to their peaks; 1 kW circulating, δ/γ = 0.58. The force follows the position but 31.8 μs late — 23° of the cycle — because the light stored in the cavity takes its decay time to adjust, and the late arrival is 31.8 μs in the formula for the delayed spring. At low frequency the delay is 2/γ(1 + δ²/γ²) = 31.9 μs. A restoring force that arrives late pushes the mirror while it is already moving back, and adds energy every cycle. — Fig. 3 One cycle of the end mirror driven at 2 kHz through $10^{-13}$ m (solid) and the restoring part of the radiation force it produces (dashed), computed by integrating the cavity’s field equation; both scaled to their peaks, with 1 kW circulating at $\delta/\gamma = 0.58$ . The force lags the position by 31.8 μs, 23° of the cycle, as the formula for the delayed spring predicts; at low frequency the lag is $2/\gamma(1+\delta^2/\gamma^2)$ = 31.9 μs.

The third figure measures the delay directly. The mirror is moved back and forth at two kilohertz, and the cavity’s field is integrated step by step as it responds. The force tracks the position faithfully in shape but arrives 31.8 microseconds behind it. At low frequencies the lag is a fixed time, $2/\gamma(1+\delta^2/\gamma^2)$ , where $\gamma$ is the rate at which the stored field decays and $\delta$ is the detuning. Close to resonance the light is stored longest and the lag is longest; far from resonance the cavity holds little and forgets quickly.

A lag of thirty microseconds on a spring whose own period is half a millisecond sounds negligible, and in one respect it is: it hardly changes the stiffness. In another it changes everything. A delayed force can be split into the part in step with the position, which is the spring, and a part in step with the velocity — a force proportional to how fast the mirror is moving. That second part is a friction term, and its sign is the whole story.

Think of a child on a swing pushed by someone who always pushes slightly late. The push is meant to bring the swing back towards the middle, but because it lags, it is still pushing back when the swing has already started to return, and so it pushes along the swing’s motion for part of every cycle. Over a cycle the late pusher does net work on the swing, and the swing goes higher each time. A restoring force that lags adds energy. On the blue side of the cavity, where the light makes a restoring spring, the delay makes that spring a source of anti-damping.

The spring that winds itself up

The same light winds one mirror up and calms another. The end mirror's motion after a small nudge, from the cavity's field and the mirror's equation integrated together, with 100 W circulating. Top: a 1-gram mirror hung as a 1 Hz pendulum with the laser on the blue side, δ/γ = 0.58. The light holds it at 625 Hz (the static spring alone gives 642 Hz) and the swing grows by a factor e every 4.0 ms, against 3.9 ms from the delayed spring; the trace is scaled to its largest swing. Bottom: the same mirror on a stiff 2 kHz mount with the laser on the red side: the light softens it to 1905 Hz and the ring dies away by e every 4.0 ms (4.0 ms predicted), where without the light — no other damping is included — it would ring on undiminished. — Fig. 4 The mirror’s motion after a small nudge, from the cavity’s field and the mirror’s equation of motion integrated together, with 100 W circulating. Top: a 1-gram mirror hung as a 1 Hz pendulum, laser on the blue side: the light holds it at 625 Hz (the static spring alone gives 642 Hz) and the swing grows by e every 4.0 ms, against 3.9 ms from the delayed spring; scaled to its largest swing. Bottom: the same mirror on a stiff 2 kHz mount with the laser on the red side: softened to 1,905 Hz, its ringing dies by e every 4.0 ms.

The fourth figure lets the mirror go. The top trace is a 1-gram mirror hung as a soft pendulum, held almost entirely by the optical spring from 100 watts on the blue side. It oscillates at 625 hertz, a little below the 642 hertz the static stiffness alone would give because the delay also softens the spring slightly at this frequency, and the swing grows by a factor of e every four milliseconds. Within a twentieth of a second a thermal jiggle of a femtometre has become a swing large enough to knock the cavity off resonance altogether. The simulated growth agrees with the complex spring — the stiffness written as a function of frequency with the lag included — to within three per cent.

This is the reason a single optical spring cannot be used on its own. It is stiff and unstable, and the instability is fast. Every laboratory that has made one has had to add damping from somewhere: an electronic servo that measures the mirror’s velocity and pushes against it, or something cleverer.

The same mechanism, multiplied, has troubled the largest optical cavities ever built. The arms of the gravitational-wave detectors whose readout what the instrument actually hears took apart store hundreds of kilowatts on mirrors of forty kilograms. A mirror has acoustic modes of its own — the whole block of glass ringing at tens of kilohertz — and light scattered by such a vibration into a neighbouring mode of the cavity can play the part of the late push, feeding the vibration from the stored power. Braginsky and his colleagues predicted this parametric instability in 2001; the detectors observed it in 2015, as a mirror mode near 15 kilohertz ringing up slowly until the interferometer lost lock, and they now suppress it deliberately. The swing that is pumped, not pushed found the same arithmetic in a child pumping a swing by standing and squatting: energy flows into an oscillation from a source modulated at the right phase.

The light that calms

The red side reverses both signs. Moving the mirror out raises the power, so the static spring is negative; and a negative spring that lags pushes against the mirror’s velocity, so the friction term is positive. Red-detuned light softens a spring and damps it.

The bottom trace of the fourth figure shows the useful version. The mirror sits on a stiff mount, a two-kilohertz spring of its own, so that the light’s anti-spring only softens it slightly, to 1,905 hertz. But the delay adds damping: a ring that would otherwise last as long as the mount’s own losses allow dies by a factor of e every four milliseconds.

Damping by light is cooling by light. A mirror in thermal equilibrium with its mount jiggles with an energy of $kT$ in each mode, and a damping force that adds no noise of its own removes that energy faster than the mount can resupply it. The mirror’s motion then corresponds to a lower temperature, in the ratio of the mount’s own damping to the total. It is the cavity version of the friction made of light, where an atom moving into a red-detuned beam scattered more photons and was slowed, and the deep reason is the same in both: the light’s response depends on motion, and on the red side the dependence opposes it. In 2011 two groups used it, one with light in a cavity around a nanometre-scale silicon beam and one with microwaves on a drumhead of aluminium, to cool a mechanical oscillator to its quantum ground state, the state in which it is jiggling only with the zero-point motion that the motion that cannot be stopped showed no bound system can lose.

The cooling has a floor, and the floor is set by the same light. Each photon reflected from the mirror delivers a random kick as well as the average push, and these kicks heat the mirror. The noise pushed below the floor found this radiation-pressure noise limiting the gravitational-wave detectors at low frequencies, where light’s graininess shakes their mirrors. For cooling, the balance between the damping the light provides and the heating its graininess causes sets the lowest occupation the mirror can reach: a fraction of a quantum when the mirror’s frequency is well above the cavity’s linewidth, and many quanta when it is well below, as it is for the one-gram mirror here.

Two beams that hold a mirror still

One beam on the blue side gives stiffness with anti-damping; one on the red side gives damping with anti-stiffness. The obvious idea is to use both, and the delay is what makes it work.

Two beams that make a spring which holds still. A 1-gram mirror held only by light: 1 kW circulating 3 half-linewidths above resonance, and a second beam 0.5 half-linewidths below it, against the second beam's power as a fraction of the first's. Solid: the frequency of the spring the two make together, in hertz; dashed: the rate at which a swing dies away, per second, negative where it grows. The first beam alone holds the mirror at 617 Hz and winds it up at 32 per second. The second is an anti-spring, but its light is stored 8 times longer, so for its stiffness it damps 8 times harder. Damping wins from a power ratio of 0.012; the spring survives until 0.094. Between them (shaded) the mirror is held stiffly and stably — at a ratio of 0.033, 496 Hz with each swing dying by e in 17 ms. — Fig. 5 A 1-gram mirror held only by light: 1 kW circulating 3 half-linewidths above resonance, plus a second beam 0.5 half-linewidths below it, against the second beam’s power as a fraction of the first’s. Solid: the frequency of the combined spring; dashed: the rate at which a swing dies away, negative where it grows. The stiff beam alone gives 617 Hz and winds the mirror up at 32 per second. Damping wins above a ratio of 0.012 and the spring survives to 0.094; in between (shaded) the mirror is held stably, at 0.033 at 496 Hz, each swing dying by e in 17 ms.

The trick is to put the two beams at different detunings. A beam far up the blue side, three half-linewidths from resonance, is stored only briefly — its light is forgotten quickly — so for a given stiffness it adds little anti-damping. A beam just below resonance is stored eight times longer, so for a given anti-stiffness it adds eight times as much damping. Mix them with the red beam’s power between about one and nine per cent of the blue’s, and the stiffness of the first outweighs the softening of the second while the damping of the second outweighs the anti-damping of the first. The last figure finds the window and checks it against the full complex springs: at a power ratio of 0.033 the mirror is held at 496 hertz, and a disturbance dies away by a factor of e in seventeen milliseconds.

Thomas Corbitt and colleagues at MIT built exactly this in 2007 and used it to hold a 1-gram mirror with no mechanical spring worth the name: the mirror was trapped by two laser beams, stiffly and stably, at a frequency of the order of a kilohertz. The point of the exercise was a mirror isolated from its surroundings, since a spring made of light carries none of the thermal agitation of a material one, and a gram-scale mirror isolated well enough is a candidate for showing quantum behaviour in an object that can be held in the hand.

The same flaw in a governor and a singing tube

The structure of this argument — a force that would be stabilising, made destabilising by a delay — runs through physics and engineering, and two earlier essays met it under different names. The governor that needed friction to work was Watt’s centrifugal regulator: the spinning balls correct the engine’s speed, but because they respond late, the correction overshoots, and Maxwell showed in 1868 that without enough friction in the mechanism the engine hunts in growing oscillations. The friction there plays the part of the red beam here. The tube that sings when heated low down was Rayleigh’s criterion for the Rijke tube: heat given to air at the moment it is most compressed adds energy to the sound, and the tube howls. A heat source that responds with the right delay to the air’s motion is a late pusher on the swing.

What the cavity adds is cleanliness. The delay is the storage time of light, known to a few per cent from the cavity’s geometry; the spring is radiation pressure, exactly $2P/c$ ; there is no friction to estimate and no flame to model. The complex spring follows from two lines of algebra, and the simulation drawn in the figures confirms it without any adjustable parameter. It is the textbook case of delay-induced instability, made by the simplest possible means.

Where the delayed spring stops being exact

The formula for the spring treats the cavity as having one mode and the mirror as moving rigidly along the axis. A real mirror tilts as well as translates, and the radiation pressure of a beam that is not perfectly centred exerts a torque. The torques make angular springs of their own. In a cavity of two curved mirrors they combine into a stiff tilt and a soft one, and once the stored power is high enough the soft one turns negative whatever the detuning — an instability John Sidles and Daniel Sigg worked out in 2006, as the detectors were being prepared for the powers that would make it matter.

The formula also assumes the motion is small: the mirror stays within a fraction of the linewidth of its operating point, so that the Lorentzian can be replaced by its tangent. The swing in the fourth figure grows past that within fifty milliseconds, and then the cavity’s response becomes nonlinear — the mirror falls out of resonance and the push collapses, or it locks into a large stable oscillation whose amplitude is set by the curvature of the peak. Optical cavities with movable mirrors show that bistability and self-oscillation, and they are well described by keeping the full Lorentzian, but not by the linear spring.

What the delay drawn here leaves out

The figures show the average radiation force, smooth and continuous, and the light that produces it is made of photons. Each reflection transfers a definite momentum at a random moment, so the force fluctuates about its average, and those fluctuations are what heat the mirror and set the floor on cooling. No figure shows them. Nor do the figures show what the mirror does to the light: a moving mirror shifts the phase of the light it reflects, so the cavity’s output carries a record of the motion, and a record of that kind is precisely what a measurement of the mirror’s position reads. The quantum limit on such a measurement comes from the trade between the noise in that record and the random kicks of the force, and its full treatment needs the noise the averages here leave out.

Still open: how massive an object can be put in a quantum state

Light has now cooled the motion of objects from nanograms to the forty-kilogram mirrors of the gravitational-wave detectors, four of which, moving together, have been brought to within about ten quanta of the ground state of their joint motion. What has not been done is to prepare a large object in a state with no classical counterpart — a superposition of two positions, or an entanglement between its motion and the light — and to show that it survives. Whether such states of massive objects behave exactly as quantum mechanics predicts, or whether some new effect, gravitational or otherwise, destroys them faster than ordinary disturbances do, is one of the questions these experiments are built to answer, and none has yet reached the sensitivity at which the proposed new effects would show.

The habit worth carrying away is to ask when a force arrives, not only how strong it is. A cavity turns radiation pressure into a spring — $1.6 \times 10^5$ newtons per metre from a kilowatt in a metre-long cavity, four-fifths as stiff as a steel wire — and because stored light takes $2/\gamma(1+\delta^2/\gamma^2)$ to respond, 32 microseconds here, the same spring winds a mirror up on one side of the resonance and calms it on the other. Any restoring force that responds late is a source of energy, and any opposing force that responds late is a source of damping; the sign of the delay’s effect is decided by the sign of the force, not by how long the delay is.