The phases that turn a glow into pulses

A laser is a pair of mirrors with something between them that amplifies light. Light that goes round the cavity and comes back in step with itself reinforces; light that comes back out of step cancels. So the cavity supports only frequencies at which a whole number of wavelengths fits into the round trip, and those frequencies are evenly spaced: the spacing is one over the round-trip time, which for a cavity a metre and a half long is a hundred megahertz. The amplifying medium can usually supply gain over a range of frequencies thousands or millions of times wider than that spacing, so a laser left to itself oscillates on many of these frequencies at once. They are called the cavity’s modes, and a laser running on N of them emits N waves superposed.

When two waves meet, they simply add, and what a detector reads is the square of the sum. The question this essay is about is what the square of the sum of N such waves looks like in time, and the answer is that it depends almost entirely on something the laser’s spectrum does not record: the phases of the modes relative to one another. With the phases equal, the output is a train of pulses of astonishing brightness. With them random, it is a glow. Everything else — the frequencies, the amplitudes, the power — is identical.

Two modes are beats, and every extra mode sharpens them #

The smallest case is already familiar. Two waves at neighbouring frequencies drift in and out of step with each other at the difference frequency, and their sum swells and fades — the beats of two tuning forks slightly out of tune. With equal amplitudes, the intensity peaks at four times one wave’s, when the two are in step, and falls to zero half a beat later, when they are exactly opposed. For two cavity modes the beat period is the round-trip time.

Add a third mode, at the next frequency up, and something new happens. All three are in step at the start, so the intensity is nine. A little later the first two have drifted apart by some angle, the second and third by the same angle, and the first and third by twice it — so the three are out of step with one another in a way that no two waves can be, and the sum falls faster than beats do. The peak is taller and narrower, and between peaks there is a small ripple where the three partly reinforce at a moment none of them is in step with all the others.

The pattern repeats exactly once per round trip, whatever the number of modes, and the reason is the spacing. Every mode’s frequency is a whole multiple of the spacing plus a common offset, so after one round-trip time every mode has advanced by a whole number of cycles relative to the others, and their relative phases are what they were at the start. The common offset shifts every mode’s phase by the same amount, which the square of the sum does not see. So the intensity is periodic in the round trip, and within each round trip its shape is decided by how the phases are arranged.

The sum can be done in closed form. Adding N unit waves whose phases advance in equal steps is summing a geometric series, and the intensity is

$$I(t) = \frac{\sin^2(N\pi t/T)}{\sin^2(\pi t/T)},$$

with $T$ the round-trip time. At $t = 0$ the ratio tends to $N^2$, and it first returns to zero at $t = T/N$. This is the same expression that describes light from a grating with N slits, and for the same reason: a grating adds N waves whose phases advance in equal steps across its width, while a laser adds N waves whose phases advance in equal steps across its spectrum. The figure is computed by adding the waves rather than by drawing the formula, and the two agree exactly at the peak.

Twenty modes in step #

The effect grows quickly with N.

With twenty modes in step the output is dark for most of each round trip and blazes briefly once per trip, at four hundred times one mode’s intensity. The pulse’s duration is set by how long the modes stay close to in step after coinciding. The fastest and slowest modes differ in frequency by N times the spacing, so they drift a whole cycle apart in one Nth of the round trip, and once they have, the modes between them are spread evenly round the circle and cancel. The pulse’s width is therefore one Nth of the round trip, whatever the round trip is.

The physical picture inside the cavity is simpler still. The pulse is a short packet of light bouncing between the mirrors. Each time it reaches the output mirror a fraction leaks out, so the laser emits one pulse per round trip. The modes are the Fourier description of that bouncing packet, and the statement that they are locked in phase is the statement that the packet exists. The two descriptions are the same thing written in time and in frequency, which is what sharpness has to be paid for says in general: a pulse short in time must be broad in frequency.

The average is the quantity that makes this worth drawing. Averaged over a round trip, the intensity is N — the sum of the N modes’ individual intensities — for the twenty-mode sum exactly as for the five-mode one. Nothing about locking the phases added energy. What changed is where in time the energy sits. The peak is N times the average, and the pulses occupy one Nth of the time. A laser that emits a watt on average through twenty locked modes emits it as pulses of twenty watts; one that emits a watt through a million locked modes emits it as pulses of a megawatt.

The same modes at random #

The same twenty waves with no relation between their phases produce something entirely different.

The average is unchanged, and the reason is the one the energy of added waves turned on. The square of a sum of N waves is the sum of their N individual squares plus cross terms, one for every pair. Each cross term oscillates at the difference between two mode frequencies, which is a whole multiple of the spacing, so over a round trip it averages to zero — whatever its phase. The phases decide what the cross terms do at each instant and cannot change their average. That is why the average intensity is N in both figures, and it is the same reason two lamps never interfere on average: the cross terms are there at every instant and cancel over time.

What the random phases do is scatter the instant-by-instant pattern. At any one moment the N waves point in N unrelated directions, the sum is a random walk of N unit steps, and its squared length is distributed exponentially with mean N. So the output is noise, with fluctuations as large as the mean, on a timescale of one Nth of the round trip. Most of the time it sits near or below the average; occasionally it spikes to several times the average, when by chance many modes happen to line up. In a round trip containing N independent intervals, the largest such spike is typically a few times N, growing with the logarithm of N — about ninety for the twenty modes drawn, not four hundred.

A laser running freely on many modes behaves like this, and on a slow detector it looks like a steady beam. The fluctuations are there, at a timescale set by the inverse of the bandwidth, and are measured by the correlation between the intensity at one instant and the next — the quantity that separates a laser from a lamp in the correlation that survives what the phase does not. But a spectrum analyser cannot tell the locked laser from the free-running one: it measures the power in each mode, and the power in each mode is the same.

The square of N at the peak, one over N in width #

On logarithmic axes the two scalings are straight lines.

The upper line has slope two because a locked peak is N waves added in step, and its amplitude is N, so its intensity is $N^2$. The lower line has slope minus one because the width is one Nth of the round trip. The product of peak and width is therefore proportional to N, which is the average intensity times the round trip: the energy in one pulse is the energy the free-running laser would have emitted over the whole round trip, delivered in a sliver of it.

The constant 0.886 is the width at half maximum of the pulse shape above, in units of $T/N$, and it belongs to modes of equal amplitude. A real laser’s modes are not equal: gain is strongest at the centre of the band and falls off at the edges, so the spectrum has a smooth envelope, and the pulse’s shape is the Fourier transform of that envelope. A spectrum shaped like a bell curve gives a bell-shaped pulse with no ripples between pulses. The product of the pulse’s duration and its spectral width takes a different constant for each shape, and cannot be made smaller than about a half: that floor is the time-bandwidth relation, and a pulse at it is called transform-limited, because nothing about the phases is wasting any of the spectrum.

The scale of real devices is where the numbers become extraordinary. A titanium-doped sapphire crystal amplifies over roughly a hundred terahertz, from about 700 to 1,000 nanometres. A cavity a metre and a half long spaces its modes a hundred megahertz apart. So the gain band holds about a million modes, and locked they give pulses about ten femtoseconds long, a hundred million times a second. The peak intensity is a million times the average. That is why ultrafast lasers are also the most intense sources in physics — intense enough to probe the vacuum’s own departure from adding exactly — and it is entirely a matter of arranging phases.

The dashed line in the figure is the random-phase comparison, and it is the reason locking is worth the trouble. The free-running laser also has occasional spikes, but they grow roughly in proportion to N rather than to $N^2$; by 256 modes the locked peak is 65,536 and the tallest random spike in a round trip about two and a half thousand. The locked laser’s advantage in peak intensity grows in proportion to N, so for a million modes it is enormous.

What locks them #

Nothing in the cavity prefers the modes in step unless something is put there to prefer it. Two approaches, both older than their best-known applications, make the pulse cheaper than the glow.

The first is active. A modulator inside the cavity opens and closes once per round trip. Light that passes it while it is open survives and light that arrives while it is closed is lost, so the cavity favours a single packet timed to arrive when the modulator is open. In the frequency description, a modulator at the mode spacing puts sidebands on each mode that land on its neighbours, coupling their phases together until they lock. This was the first method to work, in 1964, on a helium–neon laser with an acoustic modulator, and it produced pulses a few nanoseconds long.

The second is passive: something in the cavity whose loss falls as the intensity rises. A dye that absorbs weak light but bleaches under strong light, or a crystal whose refractive index rises with intensity and so focuses the brightest light more tightly through an aperture, penalises the glow and rewards the pulse. A random fluctuation that happens to be a little brighter than average loses a little less on each pass, grows at the expense of the rest, and within thousands of round trips has taken over. The second mechanism, the intensity-dependent focusing, was found by accident in 1991 in a titanium–sapphire laser that began producing femtosecond pulses when its mount was tapped, and it underlies most ultrafast lasers since. It is a nonlinearity used on purpose: the laser works because light in the crystal does not add exactly.

A ramp moves the pulse, a curve spreads it #

Given that the phases decide the shape, it matters which changes to them count.

A phase that changes by the same amount from each mode to the next — a phase proportional to frequency — is a delay and nothing else. Delaying a wave of angular frequency $\omega$ by a time $\tau$ multiplies it by $e^{-i\omega\tau}$, so a phase that falls linearly with frequency is exactly what every mode would acquire if the whole pulse were delayed. Whatever arranges the modes’ phases in a straight line against frequency has not altered the pulse at all; it has moved it. That is why the slope of phase against frequency is called the group delay, and why the packet travels at another speed from its component waves: the group velocity is the rate at which that slope builds up with distance.

A phase that curves with frequency is different, because it delays different frequencies by different amounts. With the phase rising as the square of the mode number, the group delay rises linearly across the spectrum: the high frequencies arrive later than the low ones. The pulse spreads out with its frequency sweeping from low to high across it — a chirp, named after the rising whistle of a bird — and its peak falls, because the modes are never all in step at once. The spectrum is untouched. Ten modes, equal amplitudes, the same average: a detector measuring power at each frequency sees exactly what it saw before.

Glass does this to any pulse sent through it, because its refractive index depends on frequency and a curved dependence is a curved phase. A ten-femtosecond pulse passing through a centimetre of ordinary glass comes out roughly ten times longer, and ultrafast optics is largely the art of arranging an opposite curvature — with prisms, gratings or specially layered mirrors — so that the net phase is flat again when the pulse arrives where it is needed. In an optical fibre the same curvature, working against the fibre’s own nonlinearity, is what keeps a pulse two failures keep alive travelling unchanged.

The spreading is also used deliberately. A pulse intense enough to be interesting destroys the amplifier it passes through, so in 1985 Donna Strickland and Gérard Mourou stretched pulses many times over with a large curved phase, amplified them while their peak intensity was harmless, and then applied the opposite curvature to squeeze them back. Amplification multiplies every mode’s amplitude and leaves the phases where they were put, so undoing the curvature afterwards restores the short pulse at the higher energy. Chirped-pulse amplification, recognised with the Nobel prize in 2018, is how every petawatt laser is built. It is a whole technology resting on the observation in this figure: the phases can be rearranged and put back, and the pulse follows them.

The ruler inside the pulse train #

The other description of a pulse train turned out to be just as useful as the pulses.

A train of identical pulses at a fixed repetition rate has a spectrum of lines at exactly the mode frequencies: every multiple of the repetition rate, shifted by one offset common to all. For a titanium–sapphire laser that is a million lines, each at a frequency given by two numbers — the repetition rate and the offset — both of which are radio frequencies that ordinary electronics can count. So the line spacing turns an optical frequency, far too fast to count directly, into an integer times a countable one plus another countable one. This is the frequency comb, for which John Hall and Theodor Hänsch shared the Nobel prize in 2005, and it is how optical clocks are read out and spectrographs searching for planets are calibrated.

The comb is also the most direct evidence that the modes are locked. The line spacing is exactly the repetition rate only if the pulses repeat exactly, and the pulses repeat exactly only if the modes’ phases keep their relation from one round trip to the next. A laser whose modes drift in phase has a comb whose lines smear.

Measuring the offset needs one more piece of superposition. If the comb spans a factor of two in frequency, the line at mode number $n$ can be doubled in a nonlinear crystal to give a frequency of twice the offset plus $2n$ times the spacing, and compared with the comb’s own line at $2n$, which is one offset plus $2n$ times the spacing. The two differ by exactly the offset, and they beat against each other at that frequency on a photodiode. The whole instrument is waves added and squared, with the phase relations the lock supplies making every line a ruler mark.

Still open: how short and how many #

The shortest pulses made directly by locking a laser’s modes are a few femtoseconds — about two cycles of the light — and at that length the spectrum spans an octave and the idea of an envelope and a carrier inside it starts to blur. Shorter pulses, of tens of attoseconds, are made by a different route: a strong few-cycle pulse tears electrons from atoms and slams them back, and the recombination emits a comb of high harmonics which, locked in phase by the process that makes them, sum to an attosecond burst. Whether the phases of those harmonics can be controlled well enough to reach the atomic unit of time, 24 attoseconds, cleanly and reproducibly, is being pursued now.

A different question is what happens when the modes cannot be treated as independent waves. At high enough intensity the modes interact through the medium, so the phases are not merely parameters of the sum but variables with dynamics of their own. The same mathematics describes coupled oscillators pulling one another into step, and how a laser’s modes lock spontaneously — which states are stable, how fast they form, why some lasers settle into a single pulse per round trip and others into several — is still studied as a problem in the dynamics of many coupled phases rather than solved in closed form.

The habit worth carrying away is to ask what a measurement averages out. A spectrum records the power at each frequency and discards the phases, and the phases decide whether the same power arrives as a glow or as pulses a million times brighter than the average. Two lasers with identical spectra can be entirely different light.