The correction has to be faster than the air

The first rung of this ladder ended with a limit and two ways round it. The atmosphere delivers a wavefront that is flat only over patches about ten centimetres across, so a telescope larger than a patch collects patches rather than detail, and its resolution stops improving at about an arcsecond however large the mirror.

The two ways round it are to reconstruct the image afterwards, or to fix the wavefront before it reaches the detector. This rung is about the second, and about the fact that its difficulty is not in the mirror at all.

What has to be measured, and how fast #

The correction loop has three parts: a sensor that measures the incoming wavefront, a mirror that applies the conjugate shape, and a controller between them. Each has a requirement that follows from the turbulence rather than from any engineering choice.

The spatial requirement is set by the Fried parameter $r_0$. The wavefront is flat over a patch of that size, so the sensor must sample the pupil at that spacing and the mirror must have an actuator per patch. The number of actuators is therefore $(D/r_0)^2$: for an eight-metre telescope in visible light with $r_0 = 10$ cm, that is 6,400. In the infrared at 2.2 µm, where $r_0$ scales as $\lambda^{6/5}$ to 63 cm, it is 160.

A factor of forty in actuator count between the visible and the near infrared, from one exponent. That is the first reason adaptive optics happened in the infrared first.

The temporal requirement is set by how fast the pattern changes, and the pattern changes because the wind blows it across the aperture. The relevant timescale is $r_0/v$, and the loop bandwidth needed is expressed as the Greenwood frequency,

$$f_G = 0.427\,\frac{v}{r_0}.$$

For a 20 m/s wind and $r_0 = 10$ cm that is 85 hertz in the visible and 14 in the infrared. A control loop must run at several times its required bandwidth, so the sensor must read out at several hundred hertz — which means a detector with negligible read noise at that rate, and that requirement is what held the field back for a decade after the theory was complete.

There is a fourth requirement that is easy to overlook and is often the binding one in practice: the deformable mirror must have stroke as well as actuators. The overall tip and tilt of the wavefront across a large aperture corresponds to a path difference of several micrometres, which is far more than the residual higher-order terms and more than a fast, densely actuated mirror can produce. So the correction is split: a separate fast steering mirror handles tip and tilt at large amplitude, and the deformable mirror handles everything else at small amplitude. Nearly every system built has that two-stage structure, and it exists because one term in the aberration is much larger than all the others.

The patch #

The third requirement is the one that decides what can be observed at all.

Light from two stars separated on the sky traverses different columns of atmosphere. Near the ground the columns overlap almost completely; at the height of the dominant turbulent layer they are separated by $\bar h\theta$, and once that separation exceeds $r_0$ the two wavefronts are uncorrelated. The correction measured on one star is then wrong for the other.

The angle at which the correlation is lost is the isoplanatic angle,

$$\theta_0 = 0.314\,\frac{r_0}{\bar h},$$

with $\bar h$ a weighted mean height of the turbulence. For $r_0 = 10$ cm and $\bar h = 5$ km, that is 1.5 arcseconds in the visible.

One and a half arcseconds is very small. It is a hundredth of the Moon’s diameter, a thousandth of a typical wide-field camera’s field, and — critically — it is the region within which a guide star must be found.

The star that has to be there #

A wavefront sensor needs photons. To measure the wavefront over each $r_0$-sized patch at a rate of several hundred hertz to useful accuracy requires a source of about magnitude 14 or brighter, and it has to lie within an isoplanatic patch of the target.

The density of such stars is about a tenth per square arcminute at the Galactic poles, rising by a factor of thirty in the plane. A 1.5-arcsecond patch is $6\times10^{-4}$ square arcminutes. The chance of finding a guide star is therefore about six parts in a hundred thousand, which is to say that natural-guide-star adaptive optics in the visible is not a technique but a lottery.

In the infrared the patch is eight arcseconds and its area is thirty times larger, which brings the coverage to a couple of per cent. Still small — and it is the second, larger reason the infrared came first.

Making a star #

If nature does not supply a guide star, one can be manufactured. Sodium atoms deposited by micrometeorites form a layer about ten kilometres thick at an altitude of 90 kilometres, and a laser tuned to the sodium D2 line at 589 nanometres excites them into a glowing spot of about magnitude 9 — bright enough, and pointable anywhere.

That solves the availability problem and introduces two of its own.

The cone effect is geometric. A star is at infinity, so its light traverses a cylinder through the atmosphere. The laser spot is at 90 kilometres, so its light traverses a cone whose apex is at the telescope — and the cone misses the outer parts of the turbulence that the cylinder samples. For an eight-metre telescope the mismatch is significant in the visible and tolerable in the infrared; for a thirty-metre telescope it is severe at all wavelengths, which is why the extremely large telescopes are designed with several lasers whose cones together tile the cylinder.

The tilt problem is worse and is unfixable in principle. The laser beam goes up through the same atmosphere it will come down through, and it is deflected on the way up by exactly the amount its returning light will be deflected on the way down. So the spot appears at the position the telescope’s own optics send it to, whatever the atmosphere does — and the overall tip and tilt of the wavefront, which is the single largest term in the aberration, cannot be measured from it.

A natural star is still required for tilt, but only for tilt: it need only be bright enough to measure two numbers rather than several thousand, so it can be much fainter, and the angle over which tilt remains correlated — the isokinetic angle — is several times the isoplanatic angle. That raises the sky coverage to tens of per cent, which is what makes laser systems useful.

What the correction actually delivers #

The output is a Strehl ratio, and Maréchal’s approximation gives it as

$$S \approx \exp(-\sigma^2),$$

with $\sigma^2$ the residual wavefront variance in square radians. The residual has several contributions and each carries the same exponent for the same reason.

Fitting error, from having a finite number of actuators: $\sigma^2 \propto (d/r_0)^{5/3}$ with $d$ the actuator spacing.

Temporal error, from the loop running at finite speed: $\sigma^2 = (f_G/f)^{5/3}$.

Angular anisoplanatism, from the guide star being off-axis: $\sigma^2 = (\theta/\theta_0)^{5/3}$.

Measurement noise, from finite photons on the wavefront sensor.

The recurring $5/3$ is the Kolmogorov structure function’s exponent, and its appearance in all three is not a coincidence: each term is the variance of a phase difference across some separation — in space, in time or in angle — and Kolmogorov turbulence has the same power law in all of them.

Because the terms add in the exponent, the Strehl is a product of factors, and one bad term ruins the result. A system with perfect fitting and perfect timing observing a target 5 arcseconds from its guide star in the visible has $\theta/\theta_0 = 3.3$ and $\sigma^2 = 7.5$, giving a Strehl of 0.0006. The correction is worthless off-axis in a way that no other term can compensate.

What has been done with it #

The technique’s most complete result is the Galactic centre. Following individual stars around Sagittarius A* requires resolving a region a fraction of an arcsecond across, over twenty-five years, with astrometry good to a fraction of a milliarcsecond. Both the Keck and VLT programmes did it with adaptive optics, and the orbit of the star S2 — a 16-year ellipse with a pericentre passage at 120 astronomical units — is what turned a mass estimate into a measurement.

The second is exoplanet imaging. A planet beside its star at a contrast of $10^{-6}$ and a separation of half an arcsecond is unobservable unless the star’s own light is confined to a diffraction core, because uncorrected seeing spreads it over exactly the region the planet occupies. The coronagraph does the suppression and the adaptive optics makes the suppression possible, and every directly imaged planet has been found with both. Two components have been named without being described, and the choices made in each are what separate one system from another far more than the control law does.

The mirror that has to move a thousand times a second #

The deformable mirror has been treated as a component with a number of actuators, and the way that number is delivered decides what the system can do.

The classical arrangement is a small mirror in a relay behind the telescope’s focus, a few tens of centimetres across, deformed by a stack of piezoelectric actuators pushing on its back. Such a mirror can carry several thousand actuators, moves with a bandwidth of kilohertz, and has a stroke of a few micrometres.

Its cost is optical. Reimaging the pupil onto a small mirror requires several extra surfaces, and every surface adds emissivity — which matters not at all in the visible and enormously in the thermal infrared, where the instrument’s own warm optics are the background the observation is fighting.

The alternative is to make the telescope’s secondary mirror itself deformable: a thin shell held a fraction of a millimetre off a rigid reference body and pushed by voice-coil actuators through a magnetic field. That removes every extra surface, because the secondary is a mirror the light was going to hit anyway.

The engineering is unpleasant. The shell is a metre or more across and under two millimetres thick, so it has almost no stiffness of its own and is positioned entirely by feedback; the actuators must be individually servoed at kilohertz against capacitive position sensors; and the whole assembly hangs at the top of the telescope where nothing can be adjusted easily.

Several large telescopes now carry one, and the gain in the thermal infrared is a factor of several in sensitivity — which is not an improvement in resolution at all but a reduction in the background the corrected image sits on.

A third technology has taken over at the small end. Micro-electromechanical mirrors, made by the same lithography that makes silicon chips, carry thousands of actuators on a device a centimetre across, and they cost a small fraction of a piezo stack. Their stroke is a few micrometres at most, which is why they are used behind a separate tip-tilt stage rather than instead of one.

The choice between the three is a choice about where the light is allowed to go, and it is made on the background rather than on the correction.

The mirror decides what the correction costs in background; the next difficulty decides what it is worth at high contrast, and no amount of loop speed touches it.

The aberration the loop is blind to #

There is a residual error that no amount of loop speed removes, and for high-contrast work it is the limiting one.

The wavefront sensor and the science camera do not look through the same optics. Somewhere after the deformable mirror the beam is split — a dichroic, usually, sending short wavelengths to the sensor and long ones to the camera — and everything downstream of that split is seen by one and not the other.

Any aberration in the camera’s own path is therefore invisible to the sensor. The loop dutifully flattens the wavefront at the sensor, which means it deliberately introduces the conjugate of the camera’s aberration into the beam, and the science image is worse than the loop believes.

These are the non-common-path errors, and they are static or slowly varying rather than atmospheric — a few tens of nanometres from imperfect surfaces, thermal drift and gravity flexure as the telescope tracks. That is negligible for imaging and fatal for a coronagraph, because a static wavefront error produces a static speckle in the focal plane that looks exactly like a companion and does not average away with exposure time.

The remedy is to measure the wavefront at the science focal plane itself, which removes the split. Doing so is awkward because a focal-plane image gives the intensity and not the phase, and recovering a phase from an intensity requires either a diversity — two images at different focus positions, which breaks the degeneracy — or a deliberate probe pattern applied by the deformable mirror and detected in the resulting speckle.

Both are used, and both run slowly compared with the atmospheric loop: the errors they correct drift on minutes rather than milliseconds, so a slow secondary loop running alongside the fast one is enough.

The fast loop fixes the atmosphere and a slow one fixes the telescope, and which of the two limits a given observation depends entirely on whether the target is bright and extended or faint and next to something bright.

Where the model stops #

Four limits.

The turbulence is treated as Kolmogorov and frozen — a single statistical description, blown across the aperture unchanged by a single wind. Real turbulence has several layers at different heights moving at different speeds, an outer scale beyond which the Kolmogorov law fails, and a boundary layer near the ground that behaves differently from the free atmosphere. Measuring the profile is a discipline of its own, and multi-conjugate systems that place several deformable mirrors at different conjugate heights exist precisely because the single-layer assumption is inadequate.

The correction is at one wavelength and is measured at another. Sensing in the visible and correcting in the infrared is standard practice and it works because the wavefront error in path length is achromatic — but the atmosphere’s dispersion is not quite zero, and the residual chromatic term becomes the limit for the highest-precision work.

The Strehl approximation is valid for small residuals, and it is routinely quoted where the residual is not small. At $\sigma^2 = 1$ the exponential form is a fiction, and the honest statement is the fraction of encircled energy rather than a Strehl at all.

And none of this exists in space, which is the fifth way round the problem and the one that removed the requirement rather than meeting it. One property of the whole apparatus is worth stating plainly, because it separates adaptive optics from every other way of beating the atmosphere. The correction is applied before the light is detected, so what the instrument records is already sharp: there is no deconvolution, no reconstruction, no statistical recovery of a signal from a blurred one. That is why the technique delivers spectroscopy and photometry at the diffraction limit rather than images alone, and it is why an eight-metre telescope with a working loop is a different instrument from an eight-metre telescope with a good algorithm behind it.

One more reading covers the wavelength at which adaptive optics first became routine.

Where this ladder goes next #

This rung establishes the three limits and the reason the infrared came first.

Above it lies wide-field correction: multi-conjugate adaptive optics, which uses several guide stars and several deformable mirrors conjugated to different heights to correct a field far larger than one isoplanatic patch, and ground-layer correction, which corrects only the boundary layer and improves a whole field modestly rather than a patch enormously.

Beside it lies the extreme end: the very-high-order systems built for exoplanet imaging, running at kilohertz with thousands of actuators, whose limiting error is no longer the atmosphere but the calibration of the instrument’s own non-common-path aberrations.

And below it, as the reason any of this is worth the trouble: the diffraction limit of a thirty-metre telescope at 2.2 µm is 15 milliarcseconds. Without correction, that instrument resolves no better than a ten-centimetre one, and the entire scientific case for building it rests on the three numbers in this essay coming out favourably.