A phase that survives what corrupts it

The rung below built an image out of measured Fourier components of the sky: each pair of telescopes samples one spatial frequency, the Earth’s rotation sweeps each baseline through the transform plane, and enough samples make a picture.

It left a hole in the argument. A Fourier component is a complex number, with an amplitude and a phase, and the phase is where almost all the structural information lives — an image reconstructed from correct amplitudes and random phases is noise, and one reconstructed from wrong amplitudes and correct phases is recognisable. The phase is also the part the atmosphere destroys.

Where the corruption comes from #

A signal arriving at antenna $i$ has passed through a column of atmosphere, been amplified by that antenna’s electronics, and been timed against that station’s clock. Each of those contributes a phase, and the sum of them, $\theta_i$, is unknown and changes on a timescale of seconds.

When the signals from antennas $i$ and $j$ are correlated, the measured phase is

$$\phi^{\text{meas}}_{ij} = \phi_{ij} + \theta_i - \theta_j,$$

where $\phi_{ij}$ is the phase the source actually has at that spatial frequency. The nuisance terms enter antisymmetrically and one per antenna, and that structure is the whole of what follows.

At radio wavelengths the corruption is modest at low frequency and severe at high; at millimetre wavelengths over intercontinental baselines it is total. In the optical it is worse still, and there the coherence time is the binding constraint: the phase changes in a few milliseconds, so the fringes have to be tracked faster than that or the signal averages to nothing.

The identity #

Take three antennas and add the three measured phases around the triangle:

$$\phi^{\text{meas}}_{ij} + \phi^{\text{meas}}_{jk} + \phi^{\text{meas}}_{ki} = (\phi_{ij} + \theta_i - \theta_j) + (\phi_{jk} + \theta_j - \theta_k) + (\phi_{ki} + \theta_k - \theta_i).$$

Every antenna term appears twice with opposite signs. What is left is

$$\Phi_{ijk} = \phi_{ij} + \phi_{jk} + \phi_{ki},$$

a quantity belonging entirely to the source, whatever the $\theta$ happen to be, however large they are, and however fast they change.

This is the closure phase, and Roger Jennison published it in 1958 for a radio-linked interferometer at Jodrell Bank. It is a small piece of algebra with a consequence out of all proportion to its difficulty.

Three properties are worth noticing.

It is exact, not statistical. Nothing is averaged and no assumption is made about the distribution of the errors.

It is not the same as a phase. A closure phase is a sum of three, and knowing all the closure phases does not determine the individual ones.

And it carries no absolute position. Translating the source on the sky multiplies each visibility by a phase linear in the baseline vector, and those three visibility phases sum to zero around any triangle. So closure phases are blind to where the source is — which is exactly the property that makes them immune to a delay error, and exactly the property that costs an image made from them any astrometry.

That last point is worth dwelling on, because it is a good example of a trade that recurs. The immunity and the blindness are the same fact. A closure phase cannot be spoiled by a per-antenna error because it cannot see anything that looks like a per-antenna error, and a source displacement looks exactly like one. There is no version of the identity that keeps the robustness and recovers the position; the two are the same degree of freedom, and an experiment has to choose. The same choice appears wherever a differential measurement is used to beat a systematic: what the difference removes, it also stops measuring.

What it costs #

The counting is the price of the immunity, and it is worth doing properly.

An array of $N$ antennas has $N(N-1)/2$ baselines and therefore that many measured phases. It carries $N-1$ unknown antenna terms — not $N$, because adding a constant to every $\theta_i$ changes nothing. So the number of independent pieces of source phase information available is

$$\frac{N(N-1)}{2} - (N-1) = \frac{(N-1)(N-2)}{2},$$

which is also the number of independent closure phases, out of the $N(N-1)(N-2)/6$ triangles that exist. The fraction of the phase information recovered is

$$\frac{(N-1)(N-2)/2}{N(N-1)/2} = \frac{N-2}{N}.$$

The same trick works on amplitudes, with four antennas instead of three. Each antenna has an unknown gain $g_i$ multiplying every baseline it contributes to, and the ratio

$$\frac{|V_{12}||V_{34}|}{|V_{13}||V_{24}|}$$

has every gain appearing once in the numerator and once in the denominator. The closure amplitude is therefore independent of the antenna gains in exactly the way the closure phase is independent of the antenna phases, and it is what allows an image to be made without any absolute flux calibration at all.

Self-calibration #

Closure quantities are robust and incomplete. The step that recovers the rest is an iteration that looks circular and is not.

Start with a rough model of the source — even a point source will do. Given the model, compute what each baseline’s phase should be, compare with what was measured, and solve for the $N-1$ antenna terms that best explain the difference. Apply those corrections to the data, make a new image, use it as the next model, and repeat.

The reason this converges rather than merely reproducing the assumed model is the counting above. There are far more measurements than antenna unknowns — for a 27-element array, 351 baseline phases against 26 unknowns — so the solution for the antenna terms is heavily overdetermined, and the closure quantities, which the corrections cannot change, hold it to the truth. Self-calibration cannot invent structure that violates the closure phases, and that is what makes it a measurement rather than a prejudice.

What it cannot recover is the absolute position, for the reason given above. An image made this way is correct in structure and floating in the sky, and tying it to a position requires observing a nearby calibrator whose position is known — phase referencing — which is a separate and much more demanding operation.

What the closures actually constrain #

It is worth being concrete about how much an image is pinned down by these quantities, because “some of the phase information” is a vague thing to build a picture on. The comparison to hold is with a filled aperture, where the phase is delivered by the optics and nothing has to be recovered — a single mirror measures the whole transform at once and an interferometer measures a handful of points in it, so the reconstruction is doing work an ordinary telescope never has to do.

Consider a nine-element array observing for eight hours. Each of the 84 triangles yields a closure phase at every integration, so a run with 480 one-minute integrations produces some forty thousand closure phases — of which, at any instant, 28 are independent, but across the run the baselines have moved through the transform plane and each instant’s set constrains a different part of it.

The constraint on an image is therefore enormous in count and structured in a particular way: it is entirely about relative structure. Two images differing by a translation are indistinguishable. Two images differing by a change in total flux are indistinguishable if only closure amplitudes are used. Everything else — the number of components, their separations, their brightness ratios, the asymmetry of a resolved surface — is constrained.

A useful way to hold it is that the closure phase of a triangle is zero for any centrosymmetric source and non-zero otherwise. A round star, a symmetric disc, a pair of equal components: all give zero. A non-zero closure phase is a detection of asymmetry, and it is a detection that no calibration error can fake, which is why the first thing an interferometrist looks at is not an image but the closure phases against baseline.

That is also the cleanest way to state what an image reconstruction is doing. The data do not determine a picture; they determine a set of constraints, and the reconstruction chooses among the pictures satisfying them by some criterion of smoothness or entropy or sparsity. Two reconstructions of the same data by different algorithms differ exactly where the data are silent, and comparing them is the only honest way to see which features are measured and which are chosen.

The quantity that is actually formed #

The closure phase was defined above as a sum of three measured phases. That is the right definition and it is not how the quantity is computed, because at low signal-to-noise a measured phase is a badly behaved thing.

A phase is an angle, and an angle extracted from a noisy complex number is biased: when the amplitude is comparable to the noise, the estimated phase is scattered over the whole circle and its mean is not the true phase. Averaging such angles ahead of combining them therefore biases the closure.

What is formed instead is the bispectrum — the product of the three complex visibilities around the triangle, $V_{12}V_{23}V_{31}$ — and the closure phase is taken as its argument. The antenna gain factors, complex numbers of the form $g_i g_j^*$ on each baseline, cancel in the triple product exactly as the phases do in the sum, so the bispectrum is a complex quantity belonging to the source.

The advantage is that the bispectrum can be averaged coherently. Adding complex numbers is unbiased however small they are, so many short integrations can be combined before any phase is extracted, and the phase is taken once at the end from a quantity with a good signal-to-noise ratio. Doing it the other way round — extracting a phase from each short integration and averaging the angles — throws away the coherence and biases the result.

That is not a technicality. For the hardest observations the individual integrations have signal-to-noise ratios near one, and the difference between averaging the bispectra and averaging the phases is the difference between a measurement and noise.

There is a residual bias even so, because the noise contributes to the triple product’s own amplitude, and the standard treatment subtracts the expected noise bias term before taking the argument. A quantity defined by an exact identity still has to be estimated, and the estimator is where the practical difficulty of interferometry mostly lives.

Optical, where it is much harder #

In the radio the antenna phase drifts over seconds and can be solved for. In the optical the coherence time is milliseconds, and the fringes have to be found and held in real time before anything can be measured at all.

The consequences are severe. Optical interferometers use few apertures — two to six — so the fraction $(N-2)/N$ is small: three telescopes give a third of the phase information, four give a half. Light must be piped through vacuum delay lines hundreds of metres long, with the path lengths matched to within a coherence length and adjusted continuously as the Earth turns. And a fringe tracker on a bright reference is usually required before the science target can be integrated at all.

What that buys, when it works, is the highest angular resolution anybody has achieved. A few hundred metres of baseline at visible wavelengths is a resolution of a milliarcsecond; the interferometers at Paranal reach tens of microarcseconds in the infrared. Closure phase was first used in the optical at Cambridge in the 1980s, and the images of stellar surfaces that followed — spots on a red supergiant, the flattening of a rapid rotator, the two components of a spectroscopic binary resolved for the first time — all rest on the same identity as the radio ones.

The honest limitation is sensitivity rather than resolution. Splitting light between apertures, piping it through delay lines and integrating for a coherence time leaves very little of it, so optical interferometry works on bright, nearby objects. It is the same trade every high-resolution technique makes: the correction has to happen faster than the atmosphere changes, and everything that goes into being fast comes out of the photon budget.

The same identity on one telescope #

Closure phase is usually presented as an interferometer’s technique, and one of its most productive applications is on a single mirror.

Put a mask over a telescope’s aperture with a set of small holes arranged so that no two pairs of holes have the same separation and orientation. Each pair then acts as a two-element interferometer with its own baseline, and because no baseline is duplicated, the measured fringe pattern separates cleanly into one visibility per hole pair. The telescope has been converted into a sparse array.

The point of doing that is not resolution — the longest baseline is the mirror’s own diameter, which the full aperture already has — but calibration. A full aperture’s image is corrupted by whatever residual wavefront error the adaptive optics has not removed, and that error is not known well enough to deconvolve. A masked aperture’s closure phases are immune to it in exactly the way an array’s are immune to the atmosphere, because a wavefront error over each hole is a per-subaperture phase and cancels round a triangle.

The cost is severe and is worth stating: a mask blocks most of the light, typically ninety per cent or more. So the technique trades photons for calibration accuracy, which is worth doing only when the limiting factor is the calibration rather than the noise — which is the case for detecting a faint companion close to a bright star, where the difficulty is entirely in distinguishing a real point source from a speckle.

The extension that removes the cost is to keep the full aperture and treat it as a redundant array, forming combinations of the measured image’s Fourier phases that are insensitive to small wavefront errors. Those combinations are not closure phases — the aperture is redundant, so the counting is different — but they are constructed on the identical principle, and they are what makes it possible to detect a companion inside the diffraction limit of the telescope observing it.

The identity does not care whether the apertures are on different continents or on one mirror, and the second case has produced companion detections at separations no direct image resolves.

The closure quantity’s usefulness is a matter of counting, and the count depends on the array, so it is worth reading the same construction for a much larger one and at a much shorter wavelength.

The image that rests on it #

The clearest demonstration is the millimetre-wavelength imaging of the shadow around a supermassive black hole. At those wavelengths, across baselines from Hawaii to Spain to the South Pole, the absolute phase is unrecoverable: the atmosphere and the hydrogen masers at each site put it beyond any hope of calibration.

What survives is the closure phases and closure amplitudes, and the published images were reconstructed from those quantities together with the visibility amplitudes. The array is small — of order eight sites — so the fraction of the phase information recovered is around three-quarters, and the transform plane is sampled in a handful of arcs rather than filled. Everything that follows about the sidelobes and the deconvolution applies with unusual force. The reconstructions were produced independently by several teams using different algorithms and different priors, precisely because an image from incomplete Fourier data is not unique and the agreement between methods is part of the evidence. And the visibility of a much smaller source, which is where the technique’s angular resolution is actually spent.

Where this ladder goes next #

This rung has established the closure identity, its counting, and what self-calibration adds.

The rung above is image reconstruction proper: given incomplete Fourier data and a set of closure quantities, what family of images is consistent with them, and how much of a published picture is data and how much is the regulariser. That question is not specific to astronomy and it is unusually sharp here, because the data are so few.

Beside it lies astrometric interferometry, which gives up the immunity deliberately: by referencing the phase to a nearby calibrator, it recovers the absolute position closure discards, and measures parallaxes and orbits at microarcsecond precision.

And below it, the habit: when a measurement is corrupted by an unknown, look for a combination the unknown cannot enter. The closure phase is the cleanest example in observational astronomy, and the same instinct produces the colour index, the line ratio, the flux ratio and half the other quantities the subject is built on.