The observed sky

An error budget added in quadrature

Adaptive optics does not deliver a resolution. It delivers a fraction of the light in the diffraction core, and that fraction is the exponential of minus a sum of squares. Five independent failures of the correction add in quadrature, the largest one decides everything, and the same hardware is useless in the visible and excellent in the infrared.

Assumes Seeing and Refraction.

16 min read 7 figures Light is the only messenger Measuring the unreachable

A telescope’s resolution is a property of its aperture, and the atmosphere destroys it. Adaptive optics restores some of it by measuring the incoming wavefront and deforming a mirror to cancel the distortion — but it never restores all of it, and the natural question is how much.

The answer is not a resolution. A corrected image has a diffraction-limited core sitting on a broad halo, and the useful figure of merit is what fraction of the light is in the core. That fraction is the Strehl ratio, and it is set by a sum of independent variances.

196 nanometres of wavefront, and a Strehl that depends entirely on the colour. Above, the error budget of an adaptive-optics system, term by term, in nanometres of residual wavefront. The terms are independent and add in quadrature, so the total is 196 nanometres and is dominated by the two largest; removing the smallest term entirely would improve it by under six per cent, which is why an optimisation programme that does not know which term is largest achieves nothing. Below, what that residual delivers: the Strehl ratio, the fraction of the light in the diffraction core, against wavelength. It is the exponential of minus the square of the residual measured in radians, and a radian is a wavelength, so the same physical error is four times smaller in phase at two microns than at half a micron. The system drawn here delivers 1 per cent of its light into the core at 550 nanometres and 73 per cent at 2200, and nothing about it changed between the two. — Fig. 1 Above, the error budget of a representative system, term by term, in nanometres of residual wavefront. The terms are independent and add in quadrature, so the total is dominated by the two largest and removing the smallest entirely would improve it by under six per cent. Below, what that residual delivers: a Strehl ratio that is a few per cent in the visible and most of the light in the infrared, with nothing about the system changed between the two.

That framing is unfamiliar to anybody who has met the seeing as a limit on resolution, where the answer is an angle. The difference is real rather than presentational: an uncorrected image is a blur with a width, and a corrected one is two things at once — a sharp core and a wide halo — so a single width describes neither.

Why a sum of squares, and why an exponential

Two facts do all the work.

Wavefront errors from independent causes add in quadrature. The residual phase after correction is a sum of contributions — what the mirror could not fit, what the servo could not keep up with, what the sensor could not see for lack of photons — and these are statistically independent, so their variances add. That is why an error budget is a list of squares rather than a list of magnitudes, and why the largest term dominates so completely: a term half the size of another contributes a quarter as much.

The Strehl ratio is the exponential of minus that variance. For small residuals the fraction of light remaining in the core is $\exp(-\sigma^2)$ , with $\sigma$ measured in radians of phase. The approximation is Maréchal’s, it is good to a few per cent up to $\sigma^2 \approx 1$ , and it is what turns a wavefront specification into an image quality.

The second fact carries the whole of the wavelength dependence, and it is worth stating slowly because it is the single most important thing about the subject. A wavefront error is a path difference, measured in nanometres, and it does not depend on the observing wavelength at all. The phase error is that path divided by the wavelength. So the same physical mirror error is four times smaller in phase at two microns than at half a micron, and the Strehl — the exponential of the square — is transformed out of all recognition.

A system with two hundred nanometres of residual delivers a Strehl of a few per cent at 550 nanometres and about seventy per cent at 2.2 microns. That is the reason adaptive optics was an infrared technique for twenty years, and it is not a statement about detectors or about the atmosphere.

104 nanometres of wavefront, and a Strehl that depends entirely on the colour. Above, the error budget of an adaptive-optics system, term by term, in nanometres of residual wavefront. The terms are independent and add in quadrature, so the total is 104 nanometres and is dominated by the two largest; removing the smallest term entirely would improve it by under six per cent, which is why an optimisation programme that does not know which term is largest achieves nothing. Below, what that residual delivers: the Strehl ratio, the fraction of the light in the diffraction core, against wavelength. It is the exponential of minus the square of the residual measured in radians, and a radian is a wavelength, so the same physical error is four times smaller in phase at two microns than at half a micron. The system drawn here delivers 25 per cent of its light into the core at 550 nanometres and 85 per cent at 1600, and nothing about it changed between the two. — Fig. 2 The same budget for a system built to work in the visible: every term reduced by about half, which requires roughly four times as many actuators, four times the loop speed and four times the light. The total falls from 196 nanometres to 105, and the Strehl at 550 nanometres rises from a few per cent to a usable fraction. The engineering cost of that factor of two is most of the difference between a general-purpose infrared system and a specialised visible one.

Resolution stops improving at 10 cm of aperture. Angular resolution against aperture at 500 nm, both logarithmic. The falling line is diffraction alone, 1.03 λ/D, which is what a telescope in vacuum delivers and has no floor. The curve is the same telescope under an atmosphere of Fried parameter r₀ = 10 cm, combining diffraction and seeing in quadrature: it follows the diffraction line while D < r₀ and then bends onto a plateau at 0.98 λ/r₀ = 1.01″. An amateur's 100 mm at 0.1 m would resolve 1.062″ above the air and delivers 1.47″ through it; a metre at 1 m would resolve 0.106″ above the air and delivers 1.02″ through it; the VLT at 8.2 m would resolve 0.013″ above the air and delivers 1.01″ through it; the ELT at 39 m would resolve 0.003″ above the air and delivers 1.01″ through it. At 39 m the atmosphere is costing a factor of 371: the aperture is 390 coherence lengths across and every one of the 152,100 patches it collects arrives with a phase of its own. What the extra aperture still buys is photons and speckles, and those two are what adaptive optics and speckle interferometry respectively spend to get the falling line back. — Fig. 3 The starting point the correction is trying to escape. Above about ten centimetres of aperture, adding more mirror buys light and no resolution at all, because the wavefront across a larger aperture is no longer flat. Everything in this essay is an accounting of how much of the difference between the two lines a real system recovers, and the honest summary is that it recovers a wavelength-dependent fraction of the light rather than the whole of the resolution.

The five terms, and what each one is a failure of

Fitting error is the part of the wavefront the deformable mirror cannot represent. A mirror with actuators spaced $d$ apart cannot correct structure finer than $d$ , and the atmosphere has structure at every scale. The residual scales as $(d/r_0)^{5/3}$ , where $r_0$ is the atmospheric coherence length, and since $r_0$ scales as $\lambda^{6/5}$ the fitting error in nanometres is nearly wavelength-independent — the mirror does not know what colour it is reflecting.

Servo lag is the part that changed between measuring and correcting. The air moves across the aperture at the wind speed, so the wavefront decorrelates on a timescale $\tau_0 = 0.31 r_0/v$ , which is a few milliseconds. A loop running at a kilohertz has a delay of a millisecond or two, and the residual scales as $(\tau/\tau_0)^{5/3}$ .

Sensor noise is the part that could not be measured because there were not enough photons. It rises as the guide star gets fainter and is the term that decides how much of the sky a system can work on.

Anisoplanatism is the part that is wrong because the guide star is not the target. The atmosphere is a stack of layers at different heights, so two lines of sight separated by an angle $\theta$ pass through different air above some altitude, and the correction measured on one does not apply to the other. The residual scales as $(\theta/\theta_0)^{5/3}$ , with $\theta_0$ the isoplanatic angle.

Calibration and non-common path is the part that is wrong because the wavefront sensor and the science camera do not see the same optics. The sensor is corrected to flat, and flat at the sensor is not flat at the detector. This term is static, it does not average away, and it is the one that limits high-contrast work.

$A patch 1.5″ wide in the visible and 8″ at 2.2 µm. The isoplanatic angle against wavelength, for r₀ = 10 cm at 500 nm, a turbulence layer at 5 km and a wind of 20 m/s. This is the angle over which one measurement of the wavefront is still valid, and it is the hardest of adaptive optics' three limits: at 1.5 arcseconds in the visible, the guide star has to be inside a patch a hundredth the size of the full Moon. Everything scales as λ^6/5 because r₀ does — measured off the curve at λ^1.200 — so the patch grows to 8 arcseconds at 2.2 µm, and its area by the square of that. With 0.1 stars per square arcminute bright enough to guide on, the fraction of sky reachable goes from 0.018 per cent to 0.5 — a factor of 28. That single curve is why the first working systems were infrared, why a laser is fired to make a star where there is none, and why the laser still does not solve it: a beam launched from the telescope wanders with the same atmosphere it is meant to measure, so it cannot sense the overall tilt, and a natural star is still needed for that.$

Fig. 4 The anisoplanatic term, measured rather than modelled: the angle over which a correction remains valid. It is about one and a half arcseconds in the visible and eight at two microns, because it scales the same way the coherence length does. Two consequences follow and both are severe. A correction is good over a patch far smaller than any interesting field of view, so a corrected image is sharp in the middle and blurred at the edges; and a natural guide star has to be found within that patch, which is why classical adaptive optics works on about one per cent of the sky.

It is worth noticing which of the five are properties of the atmosphere and which are properties of the instrument, because the distinction decides what money can buy. The fitting error, the servo lag and the anisoplanatism all scale with atmospheric parameters and can be reduced by building faster, denser and more expensive hardware — they are engineering. The sensor noise depends on the guide star and is therefore a property of the sky rather than of the system. And the calibration term is neither: it is a static error in the optics that could in principle be removed entirely and never quite is, because measuring it requires knowing the wavefront at the science detector, where there is no wavefront sensor. That last term is the one that limits the direct imaging of faint companions, where the enemy is not the width of the core but the speckles in the halo, and a static speckle is indistinguishable from a planet.

One more consequence of the five-thirds powers deserves attention, because it is what makes the subject so unforgiving. Three of the terms scale as a ratio raised to five-thirds, which is steeper than linear: halving the actuator spacing reduces the fitting error by a factor of three, and doubling the guide-star separation raises the anisoplanatic error by the same factor. Steep scalings cut both ways. They are why modest engineering improvements produce large gains when a system is close to working, and why a system that is a factor of two away from working is much further away than it looks. The requirement that the correction outrun the air is the same exponent seen in the time domain rather than the spatial one.

What the laser fixes and what it costs

The sky-coverage problem is solved by making a guide star: a laser tuned to the sodium D lines excites a layer of meteoric sodium ninety kilometres up, producing an artificial source wherever it is pointed.

That removes the requirement to find a bright star nearby, and it introduces two new terms.

The cone effect. A beacon at ninety kilometres is not at infinity, so the light returning from it samples a cone rather than a cylinder of atmosphere. The edges of the telescope’s aperture see air the beacon’s light never passed through, and the mismatch grows with aperture diameter. For an eight-metre telescope in the visible the residual is comparable to everything else in the budget; for a thirty-metre telescope a single beacon is useless, and the correction requires several lasers whose returns are combined tomographically.

Tip-tilt indeterminacy. The laser beam is refracted on the way up by the same atmosphere that refracts it on the way down, so the beacon appears fixed relative to the outgoing beam and carries no information about the overall tilt of the wavefront — which is the largest single component of the distortion. A natural star is still required for the tilt, but it can be much fainter, and it can be much further away, because tilt is coherent over a larger angle.

$152,100 speckles across the largest mirror drawn. The number of atmospheric coherence patches an aperture spans, (D/r₀)², against aperture, at r₀ = 10 cm. It is one at 10 cm, 100 at a metre and 152,100 at 39 m. Every one of them delivers a diffraction-limited image of the star, 0.003″ across for the largest aperture here, pointing somewhere slightly different — so a short exposure is a scatter of sharp dots and a long one is their average. The count is the reason both ways out of the atmosphere are expensive in the same way: adaptive optics has to sense and correct 152,100 patches faster than they change, which is a few milliseconds, and speckle interferometry has to record enough short exposures to average the 152,100-fold randomness away. Both scale as the same number, which is why a bigger telescope is harder to fix as fast as it is better once fixed.$

Fig. 5 What the correction is fighting, at the scale where the numbers come from. The uncorrected wavefront across a large aperture contains many independent patches, each producing its own diffraction-limited spot, and the image is the interference of all of them. The count of patches is the aperture area divided by the coherence area, and it is what sets how many actuators a mirror needs and how many photons a sensor requires — both scale with it, which is why adaptive optics on a very large telescope is much more than proportionally harder.

The two new terms illustrate a pattern worth naming: a fix for one term in a budget almost always adds terms of its own, and whether it is a gain depends on the arithmetic rather than on the idea. A laser guide star trades a sky-coverage limit for two wavefront errors, which is a clear gain for an eight-metre telescope in the infrared and a marginal one in the visible. The same trade run on a thirty-metre telescope is a loss with one laser and a gain with five, which is why the number of lasers on the coming generation of telescopes is a computed quantity rather than a choice.

It also explains why the technique arrived when it did. The idea was published in 1953 and the first astronomical systems ran in the late 1980s, and what changed in between was not the optics but the ability to read a detector, compute a correction and drive a mirror inside a millisecond. The same constraint appears wherever a correction has to outrun the thing it is correcting, and the atmosphere is an unusually demanding case because its timescale is set by wind speed rather than by anything that can be slowed down.

What was actually measured

The budget is not a theory; every term in it is measured, and the measurements are what the systems are commissioned against.

The total is measured from the image. Photometry of the corrected point spread function against a model gives the Strehl directly, and inverting Maréchal’s approximation gives the residual wavefront in nanometres. The number agrees with the sum of the individually measured terms, which is the check that the budget is complete — an unexplained excess means a term nobody listed.

Individual terms are measured by switching them off. Running the loop faster and extrapolating to zero delay isolates the servo lag; observing a guide star at a sequence of separations from a target measures the anisoplanatism directly and returns the predicted five-thirds power; observing progressively fainter guide stars measures the noise term.

And the wavelength scaling is verified across the same image. An instrument observing simultaneously in two bands measures two Strehls with one wavefront, and the ratio has to be what the exponential requires. It is, to a few per cent, across the range where the Maréchal approximation holds.

$Three apertures, one image. Long-exposure image profiles for 0.1 m, 1 m, 10 m apertures at 500 nm through an atmosphere of r₀ = 10 cm, each normalised to its own peak. Solid: through air. Dashed: what the same aperture would deliver in vacuum. The vacuum widths differ by a factor of 100 — 1.062″, 0.106″, 0.011″ — and the through-air widths differ by a factor of 1.451: 1.466″, 1.016″, 1.011″. The three solid curves are, for practical purposes, one curve. Each profile is drawn as a Gaussian of the correct full width at half maximum, which is an approximation to a real long-exposure profile — the wings of the true one are heavier — and the width, which is what the figure is about, is exact. What this picture cannot show is the exposure. In a millisecond the image is not this at all: it is 10,000 separate diffraction-limited speckles for the largest aperture here, each one as narrow as the dashed curve, scattered across the width of the solid one. Averaging them for a second is what produces the blur, and every technique that beats it works by not averaging.$

Fig. 6 The uncorrected image the whole apparatus exists to replace, and the shape it has. What adaptive optics produces is not this profile narrowed but a diffraction core sitting on a residual halo whose width is this profile’s — so a partially corrected image contains both, and the Strehl ratio is the statement of how the light is divided between them. That is why the Strehl and not the width is the figure of merit: the width of a partially corrected image barely changes as the correction improves, while the fraction of light in the core changes by orders of magnitude.

278 nanometres of wavefront, and a Strehl that depends entirely on the colour. Above, the error budget of an adaptive-optics system, term by term, in nanometres of residual wavefront. The terms are independent and add in quadrature, so the total is 278 nanometres and is dominated by the two largest; removing the smallest term entirely would improve it by under six per cent, which is why an optimisation programme that does not know which term is largest achieves nothing. Below, what that residual delivers: the Strehl ratio, the fraction of the light in the diffraction core, against wavelength. It is the exponential of minus the square of the residual measured in radians, and a radian is a wavelength, so the same physical error is four times smaller in phase at two microns than at half a micron. The system drawn here delivers 14 per cent of its light into the core at 1250 nanometres and 59 per cent at 2400, and nothing about it changed between the two. — Fig. 7 The same system pointed at a target with no bright star nearby: the anisoplanatic and sensor-noise terms have grown by a factor of two to three and now dominate everything else, and the total has nearly doubled. The instrument has not changed and its Strehl in the infrared has fallen from most of the light to a modest fraction. That sensitivity to what is in the field, rather than to what is in the telescope, is what a laser guide star exists to remove.

Where the picture stops

There are three, and the second is the reason very large telescopes need a different architecture.

Maréchal’s approximation fails when the correction is poor. At $\sigma^2$ greater than about one the exponential understates the Strehl, and at very low correction the concept stops being useful — there is no core to speak of, and the image is characterised by its width instead. The budget is a good tool exactly in the regime where the system is working.

The terms are not all independent. Quadrature addition assumes independence, and the servo lag and the sensor noise are anti-correlated in a specific way: running the loop faster reduces the lag and increases the noise, because there are fewer photons per frame. Optimising a system means finding the minimum of the sum, which sits at a loop speed that depends on the guide star’s brightness — so the optimum configuration is different for every target.

And the whole framework describes correction on one axis. For a thirty-metre telescope the isoplanatic patch is a small fraction of the field, and single-beacon correction is not the technique; the architectures under construction measure the atmosphere in three dimensions with several beacons and correct with several mirrors conjugated to different altitudes. The budget becomes a different list, and the dominant term becomes the tomographic error — the part of the atmosphere the beacons did not sample.

A fourth limit is worth adding because it is about what the number means to a user rather than to a builder. A Strehl ratio describes the fraction of light in the core, and two systems with the same Strehl can have quite different halos — one smooth, one full of static speckles — and be worth completely different amounts for a given science case. For photometry the Strehl is nearly the whole story; for astrometry what matters is the stability of the core’s position; for high contrast what matters is the structure of the halo at a few tenths of an arcsecond, which the Strehl does not describe at all. A single-number summary of an image is a compression, and which information it discards depends on what the image is for.

Why a budget rather than a specification

There is a general point about instruments hiding in this, and it is worth extracting.

A budget of independent variances has a very particular character: it is dominated by its largest term, it is insensitive to its smallest ones, and it therefore rewards effort applied in exactly one place. An engineering programme that improves four terms by twenty per cent each and leaves the largest untouched achieves almost nothing. One that halves the largest term achieves a great deal, until the second-largest becomes the largest and the programme has to move.

That structure is why the first question about any such system is which term dominates, and why the answer changes with the observing conditions, the target’s brightness, the wavelength and the airmass. There is no single figure of merit for an adaptive-optics system, and a Strehl ratio quoted without the conditions is not a number.

The same shape appears wherever independent errors combine. The residual wavefront that limits an interferometer is a budget of this kind; so is the noise in a photometric measurement, where the sky, the source and the detector each contribute a variance and the largest decides the exposure time. Recognising the shape is worth more than remembering any particular budget, because it says immediately where to look.

Close on the number that makes the whole subject possible, because it is not obvious that it should. The atmosphere’s coherence length in the visible at a good site is about ten centimetres and its coherence time a few milliseconds, and the number of independent patches across an eight-metre aperture is a few thousand. A system that measures and corrects a few thousand numbers a thousand times a second is a substantial machine and is not an absurd one. Had the coherence length been a centimetre — which it is at a poor site — the count would have risen a hundredfold and the technique would have been impossible for a generation. The parameters of the Earth’s atmosphere sit, by no design, just inside what electronics can keep up with. Every other correction in this collection is applied from a model; this one is measured and applied a thousand times a second, and that is only feasible because of an accident of scale.

There is one property of a quadrature budget worth stating plainly because it governs where effort should go, and it is routinely violated. When one term dominates, halving it is worth far more than eliminating any other — and when several terms are comparable, halving one buys almost nothing. A budget with terms of 100, 90 and 80 nanometres has a total of 156; removing the largest entirely leaves 120, a gain of a quarter for the elimination of a third of the budget. So the correct question is never “which term can be improved” but “is any term dominant”, and a well-balanced budget is a sign that the system has been optimised and that no cheap improvement remains. A badly balanced one is an opportunity. Read a published budget with that in mind and it says immediately whether the instrument was designed or assembled.

A last practical note. Because the delivered fraction is an exponential of the budget, the same system’s performance varies enormously with the observing wavelength: a wavefront error that is a small fraction of a wave in the infrared is a large fraction in the visible, and the exponent turns that into the difference between a working instrument and a useless one. The hardware does not change; the ratio of the error to the wavelength does.

Where the ladder goes next

The natural next rung is tomography: how several beacons at different positions are combined into a three-dimensional estimate of the atmosphere, and why that estimate degrades with the number of layers rather than with the number of beacons. The rung after it is the regime beyond correction — what happens when the residual is used rather than removed, which is what speckle and interferometric methods do with the same broken wavefront.