How accurate a mirror has to be

Softening an aperture’s edge treats the pupil as a transmission that varies across the aperture and finds that the rings belong to the edge. A pupil function has a second half. It is complex: an amplitude and a phase at every point, and the phase is the shape of the wavefront leaving the optic.

A perfect optic leaves a wavefront that is a perfect sphere converging on the focus. A real one does not, and the departure — measured in waves of the light being used — is what every optical specification is about.

Only one number about the error matters #

The striking thing in that figure is not the curve. It is that there is one curve.

Spherical aberration, coma, defocus and a high-frequency polishing ripple are four errors with nothing in common as shapes — one is a fourth power across the pupil, one an odd cubic, one a simple quadratic and one an oscillation with nine periods across the aperture. Normalised to the same root-mean-square, they cost nearly the same peak intensity.

The reason is short. The peak of the point-spread function is the squared modulus of the average of $e^{i\phi}$ over the pupil, and expanding the exponential for small $\phi$ gives

$$S \approx 1 - \left(\frac{2\pi\sigma}{\lambda}\right)^2 \approx \exp\left[-\left(\frac{2\pi\sigma}{\lambda}\right)^2\right],$$

with $\sigma$ the root-mean-square of the wavefront error. Only the second moment of the phase appears — the same reduction of a whole function to one number that a Debye-Waller factor performs on a lattice’s vibrations. Nothing about its shape survives to that order.

That is a genuinely useful piece of luck and it is the reason optical tolerancing works at all. An error budget can be added in quadrature — a bit from the primary, a bit from the secondary, a bit from the alignment, a bit from the air — because the quantity that matters adds that way, and the total Strehl follows from the total whatever the individual shapes were.

The approximation is quoted as valid down to a Strehl of about 0.5 and it is better than that here. What it stops describing is where the light went, which the shape decides entirely.

The core empties and the light does not leave #

This is the distinction the word “blur” flattens, and it matters more than the word suggests.

Stopping an aperture down makes the core wider and keeps essentially all of the light in it. Resolution falls and contrast does not.

Aberrating an aperture leaves the core almost exactly where it was and moves light out of it into a halo. Resolution barely changes and contrast collapses.

Which of the two is the worse failure depends on the measurement. Measuring the position of an isolated star is nearly unaffected by a halo, because the core is still there and still narrow and the centroid is still where it was — the same reason a resolution criterion is about a dip rather than about a width; the exposure has to be longer, and that is all. Seeing a faint companion beside a bright star is destroyed by it, because the halo sits exactly where the companion is and is a hundred times brighter than the diffraction rings it replaced.

That is why adaptive-optics systems on large telescopes are specified by Strehl ratio rather than by resolution: the resolution was never the problem. Uncorrected, a large telescope in the visible has a core of the right width sitting under a halo containing 99 per cent of the light, and the correction’s job is to put the light back.

The ripple case is the one that breaks the “only the root-mean-square matters” rule, and it breaks it in the place that rule never claimed. The peak is the same as for any other error of the same magnitude; the distribution of the scattered light is completely different. A smooth error puts the light in a halo near the core and a periodic error puts it in two spots at a known angle, and for an instrument looking for a planet at that angle the difference is everything.

Mid-spatial-frequency error is therefore specified separately from the total, and polishing processes are chosen and sequenced to avoid leaving periodic structure. A mirror can meet its root-mean-square specification and be useless for coronagraphy because of where its error is in spatial frequency rather than how much of it there is.

The mirror that was made to the wrong shape perfectly #

The sharpest demonstration that the quantity which matters is the wavefront and not the workmanship is the Hubble Space Telescope’s primary mirror, which was polished to an accuracy of about a hundredth of a wave and was the wrong shape.

The mirror was ground against a null corrector — an auxiliary optical device that makes a non-spherical surface look spherical to a test interferometer, so that the fringes go straight when the surface is right. The null corrector was assembled with one of its elements at the wrong spacing, by about 1.3 millimetres, because a measuring rod’s end cap had a chip of paint missing and the field lens was set against the wrong surface.

So the test reported a perfect mirror and the mirror had about half a wave root-mean-square of spherical aberration. Its Strehl in the visible was a few per cent: the core was still there, still the right width, and contained perhaps fifteen per cent of the light where it should have contained seventy. The rest was in a halo a full arcsecond across.

Three things about the episode are worth carrying.

The error was not a polishing error. The surface was one of the most accurate large optics ever made, against the shape it was told to make. The failure was entirely in what it was told, and no amount of care in the polishing could have found it.

Two independent tests said otherwise and were discounted. Two other null correctors, less precise, both reported spherical aberration of about the size that was present. They were set aside because the primary test instrument was the more accurate one, which is a defensible rule and was the wrong one here — a disagreement between a precise instrument and two rough ones is evidence about the precise instrument.

And the repair was to aberrate everything else to match. Nothing was done to the mirror. The instruments installed afterwards carried corrective optics with an equal and opposite spherical aberration, so that the total wavefront was flat again, and the telescope recovered its full Strehl. That is possible precisely because a wavefront error is a phase across the pupil and phases add: an error anywhere in the system can be cancelled by an equal and opposite error anywhere else in it.

Why a wavelength has to be quoted with a specification #

Two factors of two lie between a surface and a Strehl and both are easy to drop.

A surface error costs twice itself in wavefront, because light crosses the bump and comes back: a mirror high by 20 nanometres makes the returning wavefront 40 nanometres advanced. That is the same doubling an interferometer’s arm measures, read as a defect rather than as a signal. For a lens it is not two but the index difference less one, which for glass is about a half, so a transmitting surface is four times more forgiving than a reflecting one — which is why lens surfaces are made to looser tolerances than mirrors of the same performance.

And the cost in waves is the wavefront error divided by the wavelength. That is the factor spanning the figure: an optic used at five micrometres is twelve times more forgiving than the same optic in the blue.

So the specifications for a visible telescope and an infrared one of the same aperture differ by an order of magnitude in the polishing, which is most of the difference in their cost. The James Webb telescope’s segments are specified at tens of nanometres because it observes beyond two micrometres; a visible telescope of the same aperture would need single nanometres and could not be built to fold.

How an error budget is actually built #

Because only the second moment matters, a system’s tolerance is an arithmetic exercise rather than a design study, and it is worth working one through because the shape of the answer is not obvious.

Suppose a two-mirror telescope is required to be diffraction-limited at 550 nanometres — a Strehl of 0.8, which is a root-mean-square wavefront error of 39 nanometres. Every contribution adds in quadrature, so the budget might read: 20 nanometres from the primary’s figure, 15 from the secondary, 15 from the alignment between them, 10 from the mounting stresses, 10 from thermal gradients in the structure, and 15 from the residual of whatever correction is applied. Squared and summed, that is 35 nanometres, which fits inside 39 with a little to spare.

Two features of that arithmetic decide how such a system is designed.

Quadrature is forgiving of small terms and unforgiving of large ones. A contribution half the size of the total adds a quarter to the square, which is an eighth to the total; a contribution equal to the total doubles the square. So there is no point polishing the tenth-largest term and no escape from the largest one, and a budget is always dominated by two or three lines.

And the budget is a wavefront budget, not a surface budget. Each surface’s contribution is multiplied by its own factor on the way to the wavefront — two for a mirror at normal incidence, less for one at a grazing angle, about a half for a glass surface in transmission — so a lens element’s figure error is worth a quarter of a mirror’s. That is why a corrector plate at the front of a telescope may be made to a tolerance that would be unacceptable on the primary behind it.

The same arithmetic run backwards is how a specification is set. Decide what the system must achieve, allocate the square among the contributions in proportion to how expensive each is to reduce, and the tolerances fall out. It is an optimisation with one constraint and it is the whole of optical tolerancing, and it works only because the shape of each error drops out of the peak.

Strehl fails below a half, and it is one number about a whole image #

The Strehl approximation fails where it matters most. Below about 0.5 the exponential form stops being reliable, and at the Strehl of an uncorrected large telescope in the visible — a few per cent — it is meaningless. The computed curves in the first figure are exact and the dashed one is not, and they part company where the aberration is large.

Strehl is one number about a two-dimensional function. Two optics with the same Strehl can have quite different images, because the peak says nothing about the halo’s shape. Encircled energy, the modulation transfer function and the halo’s radial profile are all in use for that reason, and which is quoted depends on what the instrument is for.

And a real wavefront error is not static. For a telescope on the ground, most of the error is the atmosphere and it changes in milliseconds. The Strehl of a long exposure through the atmosphere is not the Strehl of any instantaneous wavefront; it is an average over a distribution of them, and the average of the exponential is not the exponential of the average.

Nor is the surface the only source. Scattering from microroughness at spatial frequencies too fine to be called a wavefront error at all removes light from the core and puts it into a wide halo that the phase treatment here does not describe. That is a separate specification, measured differently, and for a mirror exposed to dust it is often the thing that degrades first.

The number that is measured, and how #

A Strehl ratio is a comparison between a real image and an ideal one, and the ideal one does not exist to be photographed. How it is measured in practice is worth stating, because every method is indirect.

On the bench, an interferometer measures the wavefront directly: the optic under test returns a wave, it is combined with a reference, and the fringes give the phase across the pupil. The root-mean-square of that map goes into the formula. Nothing about the image is measured at all, which is the method’s strength — the answer is available before an image exists.

On the sky, the image is what there is. The peak of a star’s image is compared with the peak the aperture would give if perfect, computed from the aperture’s size and the wavelength and the detector’s sampling. That last requirement is the trap: a detector whose pixels are larger than the core measures a lower peak than is there, and the correction for it is the largest systematic in most published Strehl values.

And in a laboratory that cannot do either, the encircled energy is used instead — the fraction of the light inside a stated radius — which requires no ideal image and no assumption about sampling, and is not the same quantity. Converting between the two requires knowing the shape of the halo, which is the thing the Strehl deliberately says nothing about.

The three agree when everything is well behaved and part company exactly where the measurement is hard, which is a reliable sign that a quoted number should come with its method attached.

A phase across a pupil is not yet a surface #

They cannot show that a wavefront error is a surface at all. Everything here is a phase across the pupil, and getting from a mirror’s shape to that phase requires knowing the angle of incidence, the number of reflections and the geometry of the system. A figure of Strehl against error in waves is instrument-independent and a figure of Strehl against nanometres of glass is not.

Nor can they show the second dimension of the pupil. A one-dimensional transform gives the right functional form and the right coefficient in the exponential; the aberrations of a real system are functions of two pupil coordinates, described by the Zernike polynomials, and the accounting of which ones an optical design leaves behind is the subject of the whole aberration analysis a neighbouring essay is about.

And they cannot show the time. Adaptive optics is a control problem as much as an optical one: the correction has to be measured, computed and applied faster than the atmosphere changes, and the residual error is dominated by the lag rather than by anything in the figures here. A system with a perfect wavefront sensor and a perfect mirror running too slowly delivers a low Strehl for reasons no static calculation contains.

Still open: how the wavefront is measured when there is no star to measure it on #

Correcting a wavefront requires measuring it, and measuring it requires light from a point source in the direction of interest. Bright stars are not available everywhere, and the artificial ones — a laser tuned to excite sodium atoms in the upper atmosphere — a resonance line doing the work — producing a spot at ninety kilometres — solve most of the problem and not all of it.

Two things they do not supply. A laser spot is at a finite height, so the cone of atmosphere it samples is narrower than the cylinder the starlight crosses, and the outer parts of the aperture see turbulence the laser never passed through. And a laser beam goes up through the same atmosphere it comes down through, so the overall tilt it reports is zero by construction, and the image’s position has to be got from a real star somewhere nearby.

Both are worked around — multiple lasers to fill the cone, faint natural stars for the tilt — and the residual error from them is what sets the achievable Strehl on a large telescope in the visible. Whether visible-wavelength adaptive optics can reach the Strehl of 0.5 that would make a ground-based telescope competitive with a space one for some measurements is being attempted now and is not settled.

The habit worth carrying away is about what an error costs. A specification is meaningless without the scale it is compared against, and for anything wave-like that scale is a wavelength. Twenty-five nanometres is a superb mirror and a hopeless one, and which it is has nothing to do with the mirror.