A sharp image kept by throwing the rest away

A telescope on the ground resolves no better than one about ten centimetres across. The atmosphere delivers the incoming wavefront in patches of roughly that size, each patch delayed by a random amount, and a mirror larger than a patch collects many patches whose light interferes to form a shifting cloud of speckles rather than a single sharp image. The width of the patches is the Fried parameter, $r_0$, and the number of patches across a mirror, $D/r_0$, is the whole of the problem.

There are two established ways out, and they are opposites. Adaptive optics measures the wavefront and bends a mirror to undo it, hundreds of times a second, with an error budget whose every term is a failure of the correction. Interferometric methods use the speckles themselves, extracting the information the atmosphere scrambled but did not destroy. A third way asks less of both. It does not correct the atmosphere and does not decode it. It waits.

The atmosphere is not uniformly bad #

The figure is computed from turbulence, not assumed. Each exposure is a random phase screen with the statistics Kolmogorov’s theory of turbulence predicts — phase fluctuations whose power falls as the $-11/3$ power of spatial frequency, normalised to a given $r_0$ — drawn by the standard method of filtering random noise in the Fourier domain. The aperture is a disc laid across the screen. The overall tilt of the wavefront across it is removed, because tilt only moves a short exposure without blurring it, and moving an image is undone by shifting it back before adding. What is left is the phase error that does blur, and its variance gives the sharpness: a Strehl ratio, the peak brightness of the star’s image as a fraction of a perfect telescope’s, of roughly $e^{-\sigma^2}$ for a phase variance $\sigma^2$ in radians squared.

The screens can be checked against theory. For Kolmogorov turbulence the mean tilt-removed phase variance over an aperture is $0.134\,(D/r_0)^{5/3}$ radians squared, a result worked out in 1976; the simulated screens reproduce it to within two per cent at all three sizes.

The average is not what matters here; the spread is. At $D/r_0 = 4$ the mean variance is 1.4 radians squared, which would make an average image poor, but individual exposures range from almost useless to nearly perfect, and about three in ten have a phase error under one radian — images within a factor of three of the diffraction limit’s peak. At $D/r_0 = 7$ the mean variance has grown to 3.4 and the whole distribution has slid towards zero: none of six hundred exposures meets the strict criterion, although the best of them are still several times sharper than a typical one. At 10 the best are only a little better than the rest.

That is the observation behind the method, first made quantitative by David Fried in 1978. The atmosphere’s statistics include instants when, over a small enough aperture, the turbulence happens to be nearly flat, and an exposure short enough to freeze one of those instants — a few milliseconds, shorter than the time the wind takes to carry a patch across the mirror — is a diffraction-limited image obtained with no correction at all.

Why a flat instant is possible at all #

It is worth pausing on why the atmosphere should ever be nearly flat over a mirror, since the turbulence never stops. The answer is in the shape of Kolmogorov’s spectrum. Most of the phase variance across an aperture lives in the largest scales the aperture spans — the tilt and the slow curvatures — and the $-11/3$ power law puts ever less variance into each smaller scale. Once the tilt is removed, what remains is dominated by a handful of the next-largest modes, a focus-like curvature and a pair of astigmatisms, and a handful of random quantities are all small together reasonably often.

That is the same reason the probability collapses with size. A mirror four patches across has only a few significant modes left after tilt, and the chance that they are all small at once is large. A mirror ten patches across has dozens of modes with comparable variance, and the chance that dozens of independent random quantities are all small at once is the product of dozens of probabilities. The flat instants are not a peculiarity of the air; they are what a random field with most of its power in a few modes looks like when viewed through a window small enough to contain only those few.

A probability that falls as the square #

Fried computed how often such instants occur, and the answer is the most important number in the method.

The probability of a lucky exposure is approximately

$$P \approx 5.6\,\exp\!\left[-0.1557\left(\frac{D}{r_0}\right)^{2}\right],$$

valid when $D/r_0$ is more than about three and a half. It falls as the exponential of the square of the number of patches across the mirror. At $D/r_0 = 4$ it is nearly one in two; at 7, a quarter of a per cent; at 10, one in a million; at 12, one in a billion. The simulated screens follow the formula over the four decades the six-hundred-exposure samples can measure.

The square has a simple meaning. A lucky exposure needs the phase across the whole aperture to be nearly flat at once, and the number of independent patches that must all cooperate is the number of patches in the area of the mirror, which grows as $(D/r_0)^2$. The probability that that many independent things are all nearly right at once falls exponentially in their number. Doubling the aperture does not halve the chance of a lucky frame; it takes it from one in two to one in several thousand. Luck scales with area, and area defeats it quickly.

Why the red, and why two metres #

The Fried parameter is not fixed. It grows with wavelength as $\lambda^{6/5}$, because a phase delay measured in radians shrinks as the wavelength grows, so the same air produces larger patches in the red than in the blue.

On a 2.5-metre telescope with good seeing — a Fried parameter of twenty centimetres at 500 nanometres, about half an arcsecond — the mirror is sixteen patches across in the blue and two at 2.2 microns. The probability of a lucky exposure, depending on the square of that number, rises through five orders of magnitude across the optical and near-infrared. By the strict criterion one exposure in a hundred is lucky from about 890 nanometres and one in ten a little redder. In practice the method keeps the best one to ten per cent of exposures whether or not they meet the strict criterion, and the partially corrected images this gives extend usefully into the red around 700 to 800 nanometres.

The consequence for telescope size is a cliff. At 800 nanometres, keeping one frame in a hundred, the resolution improves as the diffraction limit $\lambda/D$ while lucky frames exist, reaching 0.077 arcseconds on a 2.2-metre mirror — six times sharper than the seeing and comparable to the space telescope’s resolution at that wavelength. A slightly larger mirror has no lucky frames at that rate, and its resolution jumps back to the seeing’s, where a larger mirror buys nothing. The best telescope for lucky imaging is the largest one that is still small enough to be lucky, and for red light and good sites that is two to three metres. The camera that first demonstrated this in the early 2000s did so on a telescope of 2.56 metres and delivered images at 0.08 arcseconds in the far red.

What luck costs #

The method is cheap in hardware and expensive in almost everything else.

It costs frames. At twenty frames a second, a telescope at $D/r_0 = 4$ yields more than five hundred strictly lucky frames a minute and one at 7 yields three; beyond about 7.6 it yields less than one. The exponential in the probability is an exponential in observing time.

It costs light. Keeping one frame in a hundred means the final image contains a hundredth of the photons collected, so its noise is that of an exposure a hundred times shorter. The trade is between sharpness and depth, set by the fraction kept, and for faint targets the best fraction is larger than for bright ones.

It costs a reference. Choosing which frames are sharp requires seeing the sharpness in each one, which requires a star bright enough to be measured in a few milliseconds — about fifteenth magnitude in the red for a two-metre telescope — within a few arcseconds of the target. The few arcseconds are the isoplanatic angle, the same limit that bounds adaptive optics: a frame lucky on the reference star is lucky on the target only if their light crossed the same turbulence, and away from the reference the luck decorrelates.

And it cost, until the 2000s, noise. A conventional detector adds a fixed read noise to every frame it reads, so reading a hundred frames and discarding ninety-nine paid a hundred times the read noise for one frame’s signal. Detectors that multiply each photoelectron in a gain register before reading, so that read noise becomes negligible and individual photons are counted, removed that cost and made the method practical for anything but the brightest stars.

Shift, select, add #

The procedure is simpler than either alternative. A camera records a stream of exposures of a few milliseconds each, tens to hundreds a second, for minutes. Each frame is scored for sharpness — usually by the peak brightness of a reference star, which is what a Strehl ratio measures — and ranked. The best fraction is kept. Each kept frame is shifted so that its reference star’s brightest point lands at the same place, which removes the tilt the atmosphere imposed on that frame, and the shifted frames are added.

The shifting alone is an old idea: adding short exposures after re-centring each on its brightest speckle was tried in the 1980s and recovers a diffraction-limited core sitting on a broad halo, because it removes the tilt from every frame but the higher-order blur only from those that happened to have little. Selection is what removes the halo, by keeping only the frames in which there was little higher-order blur to begin with. The two together — shift, select, add — are the whole of the method, and every refinement of it is about the score: weighting frames by their sharpness rather than keeping or discarding them, scoring each small region of a frame separately when the field is wider than the isoplanatic angle, or scoring on a reference too faint to measure in one frame by using many.

The method sits at one end of a spectrum of ways to use the same broken wavefront. Selection uses only the frames whose phase error was small and ignores the rest. Speckle interferometry uses every frame, extracting from each the high-resolution information its speckles still carry and averaging it statistically; its refinements recover phase information that the atmosphere scrambles differently in every frame but cannot remove from certain combinations. Adaptive optics prevents the scrambling in the first place. They trade in the same currency — the number of independent patches across the mirror — and lucky imaging is simply the one that spends it most wastefully and most simply.

What the site buys #

Because the probability depends on the square of $D/r_0$, the method is unusually sensitive to how good a night is. A site whose Fried parameter is 20 centimetres on a median night and 30 centimetres on its best tenth of nights has $D/r_0$ falling by a third on those nights, and the probability of a lucky frame at a given size rising by a factor of several hundred where the exponent is large. Lucky imaging is therefore a method for good nights at good sites, run opportunistically: the camera waits for the turbulence to be weak, as well as for the instants within a night when it is weaker still.

This is the same economics that decides how faint a star can be measured from the ground rather than from space: a ground-based telescope competes by collecting more light, and loses whenever the atmosphere’s blur costs more than the extra light gains. Lucky imaging is a way for a small ground-based telescope to buy back resolution with light it can afford to discard, and it succeeds exactly where a telescope is small enough, the wavelength long enough and the night good enough for the discarded light to be most of what it collected rather than all of it.

The method every planetary photographer uses #

The same logic has a far larger practitioner base than professional astronomy. Anyone who has photographed Jupiter through a backyard telescope in the last twenty years has done lucky imaging: a video of several thousand frames at a hundred frames a second, software that grades each frame’s sharpness, keeps the best few per cent, aligns them to remove the image motion, and adds them. The telescope is typically twenty to thirty centimetres across, $D/r_0$ is two or three in the red, lucky frames are common, and the results resolve Jupiter’s belts and storms at the telescope’s diffraction limit — a quarter of an arcsecond or better from a garden.

The planet is its own reference star, bright and extended, so the reference problem does not arise, and its surface is small enough to fit within the isoplanatic angle on a good night. The amateur method is the professional one with every cost made negligible, which is why it spread so quickly: the constraint the exponential imposes, that the aperture be only a few patches across, is automatically satisfied by the telescopes people own.

Luck and correction together #

The two opposite approaches combine naturally, because adaptive optics reduces the effective $D/r_0$ rather than eliminating it. A partially corrected image on a large telescope has a residual phase error that fluctuates, like the uncorrected one on a small telescope, and some fraction of partially corrected frames are much better than the average. Selecting them adds a further factor of sharpness to what the correction alone delivers, and it has been used to reach the diffraction limit of a five-metre telescope in visible light, where adaptive optics alone could not — a resolution of a few hundredths of an arcsecond, finer than any single telescope in space. The exponential still rules: it is the residual $D/r_0$ after correction that must be small, and every term in the correction’s error budget is a term the luck has to beat.

What the simulation leaves out #

The screens are single, frozen layers with the Kolmogorov spectrum at every scale; real turbulence has an outer scale of tens of metres that reduces the largest fluctuations, and several layers moving at different speeds, which change how quickly one lucky instant gives way to the next. The screens are drawn on a grid four apertures wide, which slightly underrepresents the largest scales — mostly tilt, which is removed anyway, and the agreement with the analytic variance says the deficit is small. Sharpness is measured by the Maréchal approximation, $e^{-\sigma^2}$, which is accurate only when the phase error is small; for the poor exposures that make up most of the distributions it is an index rather than a Strehl ratio. And each exposure is treated as instantaneous and noiseless, where real frames average over a few milliseconds of changing turbulence and are selected in the presence of photon noise, which makes the selection itself imperfect.

Still open: how far luck reaches from the reference #

The method’s hardest limit is not the exponential but the isoplanatic angle: a frame lucky for the reference star is lucky only nearby, and how far nearby depends on how high the turbulence was when that frame was taken. There is evidence that lucky frames are lucky over a larger field than average ones — that the instants when the air is flat over the mirror are disproportionately instants when the high-altitude turbulence was weak, which widens the corrected field. How much wider, how that depends on the site and the night, and whether frames can be chosen for field rather than for peak sharpness, are questions the observations have raised and not settled. The answer decides whether waiting for the air to be kind can image a field or only a star.