The width a perfect laser still has

How far a wave can remember found that the distance over which light can still interfere with a delayed copy of itself is fixed by the width of its spectral line, and why two lamps never interfere put numbers on it: thirty micrometres for a lamp filtered to ten nanometres, three hundred metres for a helium–neon laser whose line is a megahertz wide. The laser is the great exception in that comparison. Its line is ten million times narrower than any thermal source, and that narrowness is what holography, interferometric measurement of distance and every coherent receiver depend on.

It is natural to read the remaining megahertz as an imperfection — a laser not quite built well enough — and to suppose that a laser built perfectly would emit a single exact frequency forever. Arthur Schawlow and Charles Townes showed in 1958 that it would not. In the paper proposing that the maser could be extended to optical wavelengths, two years before anyone made one work, they calculated how narrow such a device’s line could be, and found a floor that no engineering removes. The floor is set by the same process that makes a laser possible at all, and it comes from a symmetry: nothing inside a laser cares what its phase is.

A phase that nothing pulls back #

A laser is an amplifier fed back on itself. Light bounces between two mirrors through a medium whose atoms have been driven into a state where stimulated emission outweighs absorption, and each pass amplifies the wave a little, until the amplification exactly balances what leaks out through the mirrors and is lost inside. The balance is stable in amplitude. If the field grows slightly, it drains the excited atoms faster, the gain falls, and the field shrinks back; if it falls slightly, the excited population recovers and the gain rises. The power of a laser is held by this feedback the way a thermostat holds a room’s temperature.

The phase has no such feedback. Stimulated emission copies whatever phase the field already has, so the gain amplifies the wave equally well at any phase, and if the phase is shifted by some random event there is nothing to restore it — every phase is as good as every other. A ball in the bottom of a bowl returns when pushed; a ball on a flat table stays wherever it was pushed to. The laser’s amplitude is the ball in the bowl. Its phase is the ball on the table.

And the phase is pushed, continually, by spontaneous emission. The excited atoms that supply the gain also emit photons spontaneously, in random directions and with random phases, and a small fraction of them are emitted into exactly the mode the laser is oscillating in. Such a photon is indistinguishable from the laser light thereafter and is amplified with it, but it arrived with a phase of its own. Adding one photon’s worth of amplitude at a random angle to a field containing $n$ photons rotates the field’s phase by an amount of order $1/\sqrt{n}$, in a random direction. Each kick is minute; there are millions of them a second, and they do not cancel. The phase performs a random walk — the process the walk that comes home studies on a line — and its mean square grows in proportion to the time elapsed.

The three runs above start together and drift apart. After a time $1/\pi\Delta\nu$ — the coherence time, where $\Delta\nu$ is the linewidth this walk produces — the phase has wandered by about a radian, and a measurement made then can no longer predict the phase from one made at the start. After many coherence times it has wandered by many full turns, in a direction nothing could have predicted. The amplitude, meanwhile, scarcely changes, because its kicks are undone by the gain as fast as they arrive.

The diffusion has a definite rate, and the argument that fixes it is short. Write the laser’s photon number as $n$ and its cavity’s linewidth as $\Delta\nu_c$, so that photons leak out at a rate $2\pi\Delta\nu_c$ each. In steady state the gain replaces them at that rate, and spontaneous emission into the mode runs at $n_{sp}$ times the replacement rate per photon, where $n_{sp}$, the inversion factor, is one for a perfectly inverted medium and larger when some atoms below the transition absorb. A random kick of unit amplitude at angle $\theta$ shifts the phase by $\sin\theta/\sqrt{n}$, whose mean square is $1/2n$. Multiplying the kick rate by the mean square per kick gives the diffusion of the phase, and writing $n$ in terms of the output power $P = 2\pi\Delta\nu_c\, n\, h\nu$ gives

$$\Delta\nu = \frac{\pi h\nu\,(\Delta\nu_c)^2\, n_{sp}}{P},$$

the Schawlow–Townes linewidth in its modern form, as a full width at half maximum. Each factor reads plainly. The width is proportional to $h\nu$ because the kicks come one photon at a time and a bigger photon is a bigger kick. It is inversely proportional to the power because more photons in the mode make each kick a smaller fraction of the field. And it goes as the square of the cavity linewidth, once because a leakier cavity needs more gain and so more spontaneous emission to sustain it, and once more because a leakier cavity holds fewer photons at a given output power.

From a wandering phase to a line #

A wave whose phase diffuses is not a single frequency, and its spectrum can be computed from the walk. The quantity a spectrometer responds to is the field’s autocorrelation: how much the field at one moment resembles the field a time $\tau$ later. For a phase that diffuses, the expected resemblance decays exponentially, as $e^{-\pi\Delta\nu|\tau|}$, because the phase difference over a delay $\tau$ is a Gaussian variable whose variance grows linearly in $\tau$, and the average of the cosine of a Gaussian variable is the exponential of minus half its variance. By the argument of the fringe and the spectrum are one measurement, the spectrum is the Fourier transform of that autocorrelation, and the transform of a two-sided exponential is a Lorentzian.

The dots above are not drawn from the formula. They are the averaged power spectrum of the simulated field itself — a unit amplitude with the wandering phase of the previous figure — and the Lorentzian fitted through them has the width the diffusion rate predicts. The shape matters as well as the width. A Lorentzian’s wings fall only as the inverse square of the distance from the centre, far more slowly than a Gaussian’s, so a laser line contains noticeably more light ten or a hundred linewidths from its centre than its quoted width suggests. Spectroscopists trying to see a faint line beside a laser’s own, and communications engineers packing channels close together, both meet those wings before they meet the core.

The same width is the inverse of a lifetime, in the sense of the width that is a lifetime: a Lorentzian line of width $\Delta\nu$ is what a process with a memory lasting $1/\pi\Delta\nu$ produces, whether what is being forgotten is an atom’s excitation or a laser’s phase. The difference is what is lost. An excited atom’s decay destroys its amplitude; a laser’s phase diffusion destroys only its phase, while the amplitude persists indefinitely. A laser is an oscillator that never stops, but whose clock drifts.

Millihertz and megahertz #

The formula gives very different answers for different lasers, and almost all of the difference comes from the cavity. A helium–neon laser with mirrors thirty centimetres apart and a finesse around three hundred has a cavity linewidth of about 1.6 megahertz. At 633 nanometres and a milliwatt of output, with a well-inverted gas, the Schawlow–Townes width is about 2.5 millihertz — a coherence time of two minutes and a coherence length of 38 million kilometres, a quarter of the way to the Sun.

A semiconductor laser is a different object. Its cavity is a chip of gallium arsenide or indium phosphide a third of a millimetre long, with mirrors formed by cleaving the crystal or by a grating etched along it; light makes a round trip in a few picoseconds and loses a large fraction each time, so the cavity linewidth is tens of gigahertz rather than a megahertz or two. Squared, that ratio is about ten billion, and even a semiconductor laser’s higher inversion factor and longer wavelength leave its quantum limit near a megahertz at a milliwatt. The straight lines on the logarithmic axes above are the $1/P$ law: ten times the power, a tenth of the width.

The obvious lever is the square of the cavity linewidth. Stretch the cavity and the linewidth falls quadratically, which is why a diode laser placed inside an external cavity a few centimetres long, closed by a diffraction grating instead of its own cleaved facet, has a line of order a hundred kilohertz rather than ten megahertz: the chip still supplies the gain, but the photons spend most of their time in a long, low-loss cavity that the chip only occupies a corner of. The grating also chooses the wavelength, which is why this arrangement became the standard tunable laser of atomic physics, where the friction made of light and the vapour-cell locks of the lines a warm vapour hides both need a line much narrower than an atomic transition.

The factor that a semiconductor adds #

When the linewidths of semiconductor lasers were first measured carefully, in 1981, they obeyed the $1/P$ law exactly — and were thirty to fifty times wider than the Schawlow–Townes formula said. The discrepancy was too large to blame on the uncertain parameters and too regular to blame on noise in the electronics. Charles Henry, at Bell Laboratories, explained it the following year.

In a gas laser, the gain and the refractive index of the medium are nearly independent at the centre of the line: changing how many atoms are excited changes how much the light is amplified but not how fast it travels. In a semiconductor they are tied together. The gain comes from electrons and holes in a band, and changing their density changes the material’s refractive index as well — strongly, because the gain peak sits on the steep side of an absorption edge rather than at the centre of a symmetric line. Henry characterised the coupling by a single number, $\alpha$, the ratio of the change in the real part of the refractive index to the change in its imaginary part, and found it to be around four to six in the materials of the day.

The consequence is that the amplitude kicks no longer stay out of the phase. A spontaneous photon that happens to add in phase with the field increases the amplitude, and the gain then relaxes back — but relaxing the gain means changing the carrier density, which shifts the refractive index, which shifts the frequency of the cavity for the few nanoseconds the relaxation takes, and so leaves behind a phase shift. Each kick now disturbs the phase twice, once directly and once through the recovery of the amplitude, and the second contribution is $\alpha^2$ times the first. The linewidth becomes the Schawlow–Townes value multiplied by $1 + \alpha^2$: seventeen for $\alpha = 4$, the dashed line in the previous figure, and thirty-seven for $\alpha = 6$. That is the whole of the 1981 discrepancy.

The factor has engineering consequences beyond linewidth. The same coupling makes a semiconductor laser’s frequency shift when its current is modulated — the chirp that limited how fast early fibre links could be driven, because a pulse whose frequency changes across it spreads in the fibre’s dispersion, as the wavelength a fibre does not smear describes. Quantum-well and quantum-dot lasers were developed partly to reduce $\alpha$, and a narrow-line semiconductor laser today is designed as much around that factor as around its cavity.

Measuring a line with its own past #

Measuring a linewidth of a megahertz on a wave oscillating at two hundred terahertz is not something a grating spectrometer can do; the best of them resolve a part in a million, which at that frequency is hundreds of megahertz. The standard method is heterodyne detection, the technique of the faint light a bright one makes measurable: mix the laser with a second laser on a fast photodiode and read the beat note at the difference frequency, which can be in the radio range and analysed electronically. But the beat between two lasers shows their combined width, and a reference laser much narrower than the one under test is often not available.

The delayed self-heterodyne method, introduced by Takanori Okoshi and colleagues in 1980, avoids the second laser by using the laser’s own past. The light is split; one half passes through a long optical fibre and the other through a frequency shifter that offsets it by some tens of megahertz; the two are recombined on a photodiode. If the fibre delays its half by much longer than the coherence time, the delayed light’s phase is uncorrelated with the present light’s, and the two halves behave as two independent lasers with identical lines. Their beat is then the line convolved with itself, which for a Lorentzian is a Lorentzian of exactly twice the width.

The long-delay curve above, computed from the simulated field rather than from the formula, has twice the laser’s width, as it should. The two shorter delays show what goes wrong when the fibre is too short. With a delay well inside the coherence time, the two halves still share most of their phase, and most of the beat’s power sits in a narrow central spike at the offset frequency, with the rest spread into a pedestal that carries fringes spaced at the inverse of the delay. Reading a width off that curve answers a different question: how well the phase was remembered across the delay, not how broad the line is.

The method therefore has a range. For a one-megahertz diode laser the coherence time is about three hundred nanoseconds, so a fibre delay of a few microseconds — a kilometre of fibre, at five microseconds per kilometre — is ample. For a laser of one kilohertz the same rule asks for three hundred kilometres of fibre, whose own losses and environmental noise ruin the measurement. Narrower lasers are measured by beating two independently built copies against each other, or against a reference cavity, and the narrowest of all by comparing several, which is how the measurement of a few millihertz below was made.

Why no laser reaches its limit #

The helium–neon laser’s quantum limit of a few millihertz is not a width any helium–neon laser has ever shown. A free-running one drifts by a megahertz in a fraction of a second, and the reason is mechanical. The frequency of a laser is set by the length of its cavity, an integer number of half-wavelengths fitting between the mirrors, so a fractional change in length is the same fractional change in frequency. A shift of one megahertz at 474 terahertz is two parts in a billion; on a thirty-centimetre cavity it is a change in mirror spacing of six-tenths of a nanometre, a few atomic diameters. A footstep in the next room, a draught warming the glass tube by a thousandth of a degree, or the hum of a cooling fan moves the mirrors that much, and the frequency follows.

So the linewidth a real laser has is, almost always, the record of its environment rather than of its quantum physics. That technical noise also has a different spectrum from the Schawlow–Townes walk: much of it is slow and drifting, with more power at lower frequencies, and the line it produces is roughly Gaussian rather than Lorentzian and keeps getting wider the longer it is observed. Measured linewidths are therefore quoted with an observation time, and a laser that is one kilohertz wide over a millisecond may be a megahertz wide over a minute. The quantum-limited part survives as the Lorentzian wings and as the floor on very short timescales.

Removing the technical noise is the work of laser stabilisation, which locks the laser to an external reference cavity built to be as quiet as possible: two mirrors optically contacted to a spacer of ultra-low-expansion glass or single-crystal silicon, suspended in a vacuum, isolated from vibration and held at the temperature where the material’s expansion passes through zero. The best of these, a silicon cavity cooled to 124 kelvin at the national metrology institute in Braunschweig, has produced light at 1.5 micrometres with a linewidth below ten millihertz — near the Schawlow–Townes limit of an ordinary gas laser, achieved by a very different route. Those lasers are the local oscillators of the optical clocks in the clock that measures a height, where the length of time a phase can be trusted sets how precisely a frequency can be read, as the wait that sets the precision argues for any clock.

The bars above translate linewidths into coherence lengths, $c/\pi\Delta\nu$, on an axis spanning eighteen orders of magnitude. The quantum limit of a helium–neon laser and the best stabilised laser both reach tens of millions of kilometres; a stabilised helium–neon of the kind used for length measurement, a hundred kilometres; a telecom diode laser, a hundred metres; the yellow line of a sodium lamp, two centimetres; and a white light-emitting diode, a micrometre, which is why the depth a broad spectrum can see can use such a source as a ruler and why nothing else can.

Masers, thermal photons and the factor of two #

Schawlow and Townes derived their result for the maser first, where it takes a slightly different form. At microwave frequencies a photon’s energy is tiny compared with the thermal energy at room temperature — $h\nu/k$ is a twentieth of a kelvin at the 1.4-gigahertz hydrogen line — so the cavity is already full of thermal photons, thousands per mode, and their random phases kick the oscillation far harder than spontaneous emission does. The factor $n_{sp}$ is then replaced by roughly $kT/h\nu$, and the hydrogen maser’s linewidth is limited by the temperature of its cavity rather than by the quantum discreteness of light. At optical frequencies the thermal occupation is about $10^{-33}$, and only spontaneous emission is left. The same formula covers both, with whichever source of random-phase photons is larger.

The inversion factor itself is a statement about how far the medium is from ideal. A perfectly inverted medium, with every atom excited, has $n_{sp} = 1$ and adds the minimum possible noise, which is the same minimum that the noise every amplifier must add shows any phase-insensitive amplifier must pay; a medium in which some atoms remain in the lower level absorbs as well as amplifies, needs more excited atoms to reach the same net gain, and so produces more spontaneous emission per unit of gain. The language of hotter than any temperature there is makes the connection direct: an inverted population is a negative temperature, and $n_{sp}$ is the occupation number a thermal mode would have at that temperature, with the sign reversed.

The formula has also been argued over by a factor of two. Schawlow and Townes’s original expression was for an oscillator near threshold, where the amplitude still fluctuates freely along with the phase; well above threshold the amplitude is clamped and only the phase noise contributes to the line, which halves the width. Melvin Lax and others sorted this out in the 1960s, and the modern formula is the above-threshold one. The distinction is visible in the laboratory: as a laser’s pump is raised through threshold its linewidth falls steeply, by the $1/P$ law and by an extra factor of two as the amplitude locks.

The symmetry and the phase #

The reason the laser line has a width at all is worth stating as a principle, because it reaches well beyond lasers. When a system’s state can be shifted continuously without changing its energy — here, the overall phase of the field — fluctuations along that direction are not restored, and they accumulate. A superconductor’s phase, a crystal’s position, the direction of a magnet’s magnetisation in a perfectly isotropic material and a laser’s phase all have this freedom. In each, a symmetry that the governing equations have but the state does not leads to a direction along which noise diffuses without limit. In a laser the diffusion is slow enough to be measured in millihertz, and it is the floor every other source of broadening sits on.

There is a version of the same argument in quantum mechanics itself. A field with a sharply defined photon number has an undefined phase, and one with a defined phase cannot have a defined number, by the trade-off in the state that swings like a pendulum. A laser’s output is close to a coherent state, with photon number uncertain by about its square root and phase uncertain by about one over the square root of the number. The Schawlow–Townes walk is that phase uncertainty being renewed continuously by spontaneous emission as photons leave the cavity and are replaced — which is why the formula contains Planck’s constant, and why no improvement in mirrors, glass or electronics can remove it.

A phase walk with nothing else in it #

The simulated walk treats the kicks as a smooth Gaussian process with no correlation from one step to the next, which is correct when the kicks are far faster than the coherence time and far weaker than a radian each, as they are in any laser above threshold. It omits the amplitude dynamics entirely, and with them the relaxation oscillations — a damped ringing of amplitude and inversion at a few gigahertz in a semiconductor laser — that put side peaks onto a diode’s spectrum and through $\alpha$ couple into its phase. The linewidth limits in the third figure use representative cavity parameters, not those of any particular laser, and the semiconductor curve stands for a family of devices whose cavity linewidths span a factor of several. The coherence lengths in the last figure combine a quantum limit, a record measurement and typical commercial specifications, and each of the last group could be shifted an order of magnitude by choosing a different product.

Still open: whether the limit itself can be lowered #

The Schawlow–Townes width is a floor for a laser in which the gain medium responds faster than the cavity — the usual case. When the arrangement is reversed, with atoms whose coherence lasts far longer than a photon’s stay in the cavity, the atoms rather than the cavity carry the phase memory and the limit can fall far below the cavity-based value. These are the “superradiant” or “bad-cavity” lasers, demonstrated with strontium atoms on a millihertz-wide clock transition, and their promise is an optical frequency reference that does not need a vibration-isolated reference cavity at all; how close they can come to the atoms’ own natural width, and whether they can run continuously rather than in bursts, is an active question. Separately, squeezed states can in principle reduce the phase noise of a laser below the coherent-state level, though doing so inside a working oscillator, rather than in light produced outside one, has been shown only in limited forms.

The habit worth carrying away is to ask what a system’s equations do not care about. Whatever can change without costing energy is not held in place, so noise walks it freely — and a laser’s phase, kicked by every spontaneous photon that joins the mode, walks at a rate $\pi h\nu(\Delta\nu_c)^2 n_{sp}/P$ that is 2.5 millihertz for a milliwatt helium–neon laser and a megahertz for a semiconductor one, multiplied there by $1 + \alpha^2$ because its gain and its index move together. A perfect laser still has a width, and that width is the price of having a phase at all.