The correlation that survives what the phase does not

A star is a disc of a few thousandths of a second of arc, and no telescope resolves it. Two telescopes can, by interfering their light, and doing so means holding two optical paths equal to a fraction of a wavelength while the atmosphere moves them by microns several times a second.

Michelson did it in 1920 and measured Betelgeuse. Almost nobody repeated it for forty years, because the engineering is brutal and the reward is one number per heroic night.

The alternative is to discard the phase deliberately. Point two separate telescopes at the star, record how the brightness at each fluctuates, and ask whether the fluctuations at the two are correlated. That measurement contains the star’s diameter, it needs the two paths equal only to a few metres, and it was so unexpected when it was proposed that its authors spent several years defending the arithmetic before anybody would look through the instrument.

What the two instruments measure #

The object both are after is the complex degree of coherence $\gamma(\mathbf{b})$ — the normalised correlation of the field at two points separated by a baseline $\mathbf{b}$. For a source of angular brightness distribution $I(\hat{\mathbf{s}})$ it is the Fourier transform of that distribution, which is the van Cittert–Zernike theorem and is why an interferometer measures a shape at all — the same transform relation that connects a spectrum to a coherence time, applied across the sky instead of along the beam.

The two instruments measure different things and answer the same question. One combines the light itself, so what it records is a fringe whose contrast depends on the relative phase of the two arrivals; the other records the intensity at each telescope separately and correlates the two numbers afterwards. The first is more sensitive and the second is nearly immune to anything that disturbs the phase — and since the atmosphere disturbs the phase constantly, the second works on nights the first cannot.

A Michelson interferometer measures $|\gamma|$, because the contrast of the fringes it forms is exactly that. An intensity correlation measures $|\gamma|^2$, because — for light from a thermal source — the excess correlation of the intensities at two points is

$$\frac{\langle \Delta I_1 \Delta I_2\rangle}{\langle I_1\rangle\langle I_2\rangle} = |\gamma(\mathbf{b})|^2.$$

One curve is the square of the other, so neither contains less information about the source’s size. What the squaring loses is the phase of $\gamma$, which the intensity correlation never sees — and that loss matters for imaging an asymmetric object and does not matter at all for measuring a diameter, since a diameter is read off where the curve first vanishes.

The two curves reach zero at the same baseline. That baseline is $1.22\lambda/\theta$, the same number as a telescope’s resolution limit with the aperture replaced by the separation, and finding it is the whole measurement.

The four paths, which is where the square comes from #

There is a way of seeing the square that is worth having, because it explains why the answer is $|\gamma|^2$ and not $|\gamma|$ or $|\gamma|^4$.

Take two points on the source, $a$ and $b$, and two detectors, $1$ and $2$. A joint detection — one photon at each detector — can happen two ways: $a$ to $1$ and $b$ to $2$, or $a$ to $2$ and $b$ to $1$. The final state is the same in both cases, since the photons are identical and no measurement records which emitter each came from, so the two amplitudes must be added before squaring.

Squaring a sum of two amplitudes produces a cross term, and the cross term is the product of the two phase factors, which depends on the source separation and the detector separation together. Summing that cross term over all pairs of source points gives exactly $|\gamma|^2$: two factors of $\gamma$, one from each of the two paths that were exchanged.

An ordinary interference experiment adds two amplitudes for a single photon and gets one factor of $\gamma$. The intensity correlation adds two amplitudes for a pair and gets two. That is the whole of the difference between the two instruments, stated without any statistics.

Why the intensities are correlated at all #

The result looks like a claim about photons cooperating, which is what made it contentious. It is not; it is a statement about a classical wave.

Light from a star is the sum of the fields radiated by an enormous number of independent atoms, which is exactly the situation in which two lamps cannot interfere. Adding many independent complex amplitudes with random phases gives a resultant whose amplitude wanders — a random walk in the complex plane — so the intensity fluctuates on the timescale over which the phases scramble, which is the inverse of the light’s bandwidth. That is the speckle of a thermal source in time rather than in space, and it is present in starlight as surely as in laser light scattered from a wall.

Light from many independent emitters is not smooth. Its intensity fluctuates with an exponential distribution whose standard deviation equals its mean — so a thermal field is, moment to moment, half as bright or twice as bright as its average, all the time. The correlation being measured here is a correlation between those fluctuations, and it exists only because the light is not steady.

Two detectors placed close together sample the same wandering and their fluctuations agree. Move them apart and the wandering becomes different at the two, because the phase relationship between the many emitters is different as seen from the two positions — and how quickly the agreement is lost with separation is precisely how big the source is.

Nothing more is required. The correlation is between two classical intensities, and the photons are reporting where the classical intensity happened to be large. Written in photon language the same fact reads as bunching: photons from a thermal source arrive in clumps, and the clumping is correlated between two detectors that see the same clump.

The subtlety Purcell settled in 1956, in a half-page letter, is that the excess is not an interaction. Bosons prefer to occupy the same state, and the correlation is the counting statistics of that preference — which for light of low occupancy is a small effect on top of ordinary shot noise, and which is exactly the classical wave result re-derived.

The timescale of the wandering is set by the bandwidth, and the two are reciprocal: a filter passing $\Delta\nu$ gives an intensity that stays put for about $1/\Delta\nu$. For a one-nanometre filter in the green that is a coherence time of about a picosecond, which is a thousand times shorter than any electronics can resolve — and this is the crucial practical fact of the whole subject. The correlator does not resolve the fluctuation. It integrates over many coherence times, and the correlation it recovers is diluted by the ratio of the correlator’s resolving time to the coherence time.

That dilution is not a defect to be engineered away; it is what buys the tolerance. The instrument is insensitive to path differences up to $c$ times its resolving time precisely because it has given up resolving the fluctuation, and those are the same sentence.

What it buys #

The advantage is one number and it is enormous.

What it buys is baseline. A phase-sensitive interferometer needs its two paths matched to within a fraction of a wavelength, which is why the longest such baselines are measured in hundreds of metres and require tunnels. Correlating intensities needs only that the two records be timed against each other, so the telescopes can be kilometres apart and connected by a cable — and angular resolution goes as the baseline, so that difference is the whole point of the technique.

A fringe exists only if the two paths are equal to a fraction of a wavelength. Fifty-five nanometres is the tolerance for visible light, and the atmosphere violates it continuously: the wavefront arriving at two telescopes tens of metres apart is corrugated by many microns, varying on a timescale of milliseconds. Everything difficult about optical interferometry is the fight against that.

An intensity correlation compares two slowly varying quantities — the brightness at each detector, filtered to the bandwidth of the electronics. Two arrivals count as simultaneous if they fall within the correlator’s resolving time, so the path tolerance is $c$ times that: for a hundred-megahertz correlator, three metres.

Eight orders of magnitude of tolerance is not an improvement in degree. It means the light collectors need not be optical quality at all — Hanbury Brown and Twiss used searchlight mirrors of a few metres aperture, made of segments, focused only well enough to put the star on a photomultiplier. It means the baseline can be extended by rolling one of them along a railway track. And it means the atmosphere is irrelevant, because scintillation changes the intensity at each detector on a timescale far longer than the correlation being sought and averages out of the product.

What it costs #

The correlated signal is very small, and the reason is worth stating as a number rather than as an adjective.

The excess correlation is $|\gamma|^2$ of the fluctuations, and the fluctuations are of order the mean intensity only when there is at least one photon per mode. The relevant figure of merit is the degeneracy parameter: the number of photons arriving per coherence time, which for a bright star observed through an optical filter is about $10^{-3}$.

That is the factor by which the correlated signal sits below the shot noise. Recovering it requires averaging over an enormous number of coherence times, and the signal-to-noise ratio comes out as the degeneracy parameter times the square root of the bandwidth times the integration time. With a hundred-megahertz correlator and an hour of integration that is a few hundred standard deviations — comfortable, and only because the star is bright.

Two consequences follow and both shaped the instrument’s history. Sensitivity does not improve by using a wider optical filter, because a wider filter admits more photons and shortens the coherence time in the same proportion, leaving the degeneracy parameter unchanged. And the method works on the brightest few dozen stars and on nothing else: Hanbury Brown’s Narrabri instrument measured 32 diameters between 1965 and 1972 and then stopped, having exhausted its sky.

The technique also has a form of aperture synthesis available to it that the amplitude version does not, and it comes free. Because there is no need to bring the light of the two collectors together physically, an array of $N$ collectors provides $N(N-1)/2$ baselines simultaneously, each correlated with each in electronics, with no optical combination and no delay lines. An amplitude interferometer with $N$ apertures needs a beam-combining scheme whose complexity grows at least as fast, and every path in it has to be stabilised.

The measurement that was not believed #

The reception is worth recounting because the objection was reasonable and the answer was not obvious.

Hanbury Brown and Twiss reported in 1956 that the arrival times of photons at two detectors illuminated by the same thermal source are correlated. Several physicists replied in print that this could not be so: photons are emitted independently by independent atoms, so their arrivals must be a Poisson process, and a Poisson process has no correlations. Three separate groups attempted the experiment and reported null results.

The resolution came in three parts. Purcell pointed out that photons are bosons and that the counting statistics of bosons in the same mode are not Poisson. Hanbury Brown and Twiss pointed out that their own experiment was in any case a classical one, in which the correlated quantity is a wave intensity that anybody could compute from Maxwell’s equations. And the null results turned out to have used light that was insufficiently coherent — sources whose degeneracy parameter was too small, or detectors whose bandwidth was too wide, so that the small excess was diluted below detection.

The episode is a good example of a disagreement that looked like a dispute about physics and was mostly a dispute about what the experiment was measuring. Once it was clear that the object being correlated was an intensity rather than an interference, the classical calculation was uncontroversial and the quantum one agreed with it.

The same correlation, run the other way #

Measuring $g^{(2)}$ became a standard tool, and its most-used application is the opposite of the one it was invented for.

A laser has $g^{(2)}(0) = 1$: its intensity does not fluctuate, so there is no excess correlation, and an intensity interferometer fed with laser light measures nothing about anything. That is not a failure of the technique but a statement that a coherent state has no intensity noise to correlate.

A single emitter gives $g^{(2)}(0) = 0$. Having emitted one photon it must be re-excited before it can emit another, so two photons never arrive together and the correlation at zero delay collapses instead of rising. Antibunching of this kind cannot be produced by any classical field, and it is the cleanest available evidence that light arrives in lumps, and measuring it is the standard proof that a source is a single quantum emitter — a single molecule, a single defect in diamond, a single quantum dot. Every single-photon source announced in the last thirty years is announced with this curve.

So the measurement distinguishes thermal light, laser light and single-emitter light by one number, and the three values $2$, $1$, $0$ are as clean a classification as quantum optics has.

The instrument built from the argument #

Narrabri, in New South Wales, is worth describing because the design decisions are all consequences of the paragraphs above.

Two reflectors of $6.5$ metres aperture, each made of 252 hexagonal mirrors on a steel frame, with a focus good to a few centimetres — not an optical figure, because none is needed. Each ran on a circular track $188$ metres across, so the baseline could be set to anything from $10$ to $188$ metres while both stayed pointed at the same star, and the track was circular rather than straight so that the baseline projected onto the sky stayed perpendicular to the line of sight as the Earth turned.

Each reflector fed a single photomultiplier through a narrow filter. The two photocurrents were brought together electrically, multiplied, and integrated. The interesting engineering was in the cables and the multiplier, not the optics; the interesting astronomy was in choosing which of the few dozen sufficiently bright stars to spend a hundred hours on.

The result was 32 angular diameters of hot stars, published in 1974, and they remained the calibration standard for stellar temperature scales for decades. Every one of them was obtained with two mirrors that could not have formed a usable image of anything.

Where it stops #

The source has to be thermal for the $|\gamma|^2$ relation to hold. The Siegert relation $g^{(2)} = 1 + |\gamma|^2$ is a property of Gaussian statistics, which a sum of many independent emitters has and a laser does not. Applying the technique to a source that is partly coherent requires knowing how much, which usually means the technique is the wrong one.

The baseline is a projection. What is measured is the coherence across the vector separation as seen from the source, so an east–west baseline at one hour angle is a different measurement from the same baseline six hours later. Building up a curve means either many baselines or letting the Earth rotate, and both were done at Narrabri.

The correlator’s bandwidth is the instrument’s real aperture. Everything about the sensitivity is set by how finely two arrivals can be timed, and nothing about the collectors improves it except their area. That is why the technique’s revival waited on electronics rather than on optics: a modern photon-counting detector and a nanosecond correlator recover an order of magnitude in signal-to-noise over the 1960s apparatus with the same mirrors, and the arrays of Cherenkov telescopes now used for it were built for a different purpose entirely and have kilometre baselines by accident.

And the diameter is a model. The curve is fitted with a uniform disc, and a real star is limb-darkened, so the fitted angular diameter is systematically smaller than the true one by a few per cent. That correction is theoretical rather than measured, and it is the largest systematic in every diameter obtained this way.

The same curve underlies both instruments, and that is the cleanest way to state the relationship between them. Fringe visibility from two mutually incoherent sources falls with baseline in a way fixed by the source’s angular size; one instrument measures that visibility directly and the other measures its square. Squaring throws away the sign and keeps the magnitude, which is precisely what makes the second immune to the atmosphere and blind to the phase.

A baseline plays the role an aperture plays in ordinary imaging. Two points are resolved by a single aperture when their patterns can be told apart, and an interferometer’s coherence vanishes at a baseline set by the same angular size — so a separation of telescopes buys resolution the way a wider mirror does, without anyone having to cast the glass in between.

The ladder from here #

Later rungs on this anchor: closure phase, which recovers the phase information intensity interferometry throws away by combining three apertures; the modern revival, in which arrays of atmospheric Cherenkov telescopes built for gamma-ray astronomy are used at night as intensity interferometers with kilometre baselines; photon-counting correlations of non-classical light, where the Siegert relation fails and the failure is the measurement; and the connection between the second-order coherence and the fluctuation–dissipation theorem, which is the same statistics seen from thermodynamics.

The neighbouring ladders are the fringe that measures a star, which is the amplitude instrument; the grain that is in the light, which is the same fluctuation in space; and why two lamps never interfere, where the coherence being measured here is first defined.