The focus that is two lines

In 1801 Thomas Young, looking at fine lines through his own left eye, found that he could see horizontal lines sharply at one distance and vertical lines sharply at another, and never both at once. He measured the difference and traced it to the shape of his eye’s optics, and his paper was the first description of the defect now called astigmatism. Twenty-five years later George Airy, who had the same trouble with his left eye, had a lens ground with a cylindrical surface — curved in one direction and flat in the other — and could see with it as well as with his right. Spectacle prescriptions have had three numbers in them ever since: a sphere, a cylinder, and the angle at which the cylinder is turned.

The defect is not a blur in the ordinary sense. A defocused eye or camera has a focus that is in the wrong place; an astigmatic one has no focus anywhere. Following the light through an astigmatic lens step by step shows exactly what it has instead, and why the cure has to be a lens that is not round.

A lens with two focal lengths #

What a lens is doing drew a lens as a single bending power, the same in every direction across its face. A lens ground with different curvatures in two perpendicular directions — a toric surface, shaped like a patch of a doughnut — has two: one for the rays spread out along the vertical, one for those spread along the horizontal. Light entering such a lens as a parallel beam converges faster in the stronger direction than in the weaker, and the two directions reach a focus at different distances.

Follow a ray entering the lens at a point $(X, Y)$ on its aperture. Its horizontal position is governed by the horizontal power alone, and shrinks in proportion to its distance from the horizontal focus; its vertical position likewise by the vertical power. At a distance $z$ behind the lens it crosses at

$$\left(X\left(1 - \frac{z}{f_h}\right),\; Y\left(1 - \frac{z}{f_v}\right)\right).$$

At the vertical focus, $z = f_v$, every ray has the same vertical position — zero — and the beam is a horizontal line. At the horizontal focus it is a vertical line. Between and beyond them, the beam’s cross-section is an ellipse.

The pencil of light behind an astigmatic lens is called Sturm’s conoid, after the mathematician who described its geometry in 1845. It has two waists, each of them a line, at right angles to each other, and between them a region where the cross-section turns from a flattened ellipse through a circle to an ellipse flattened the other way. There is no point anywhere. The circle is the best the beam ever does: the circle of least confusion, the smallest patch into which the lens gathers light from a point.

Halfway, in dioptres #

Where is the circle? The beam’s horizontal half-width at distance $z$ is the aperture’s half-width times $|1 - z/f_h|$, and its vertical half-width is the same with $f_v$. The cross-section is round where the two are equal, and solving gives

$$z_c = \frac{2f_vf_h}{f_v + f_h},$$

the harmonic mean of the two focal lengths. Its radius is the aperture’s times $(f_h - f_v)/(f_h + f_v)$ — a ninth of it, for focal lengths of 100 and 125 millimetres.

The harmonic mean is a little nearer the stronger focus than the arithmetic mean: 111.1 millimetres rather than 112.5. In powers it is exactly halfway: the powers of the two meridians are 10 and 8 dioptres, and the circle sits where a lens of 9 dioptres would focus. That is the natural measure. A beam’s width near focus grows in proportion to how far it is from focus measured as power, which is why opticians and optometrists work in dioptres and why “halfway between the line foci” means halfway in dioptres. The focus that is a slab, not a plane found that every lens tolerates some defocus before the blur shows, and the tolerance is a fixed number of dioptres for a given aperture; an astigmatic lens whose two powers differ by less than that tolerance has astigmatism nobody will notice.

The clock face #

An astigmatic eye sees lines differently according to their direction, and the reason is visible in the conoid. If the retina sits at the first line focus, a point of light is imaged as a short horizontal line. A horizontal line in the scene, which is a row of such points, is imaged as a row of overlapping horizontal lines — a horizontal line, sharp. A vertical line is imaged as a row of horizontal smears stacked vertically — a vertical band, blurred to the width of the smear.

That is what the clock-face chart on an optician’s wall measures. It is a set of black spokes radiating at every angle. An eye with no astigmatism sees them all equally black; an astigmatic eye, focused at or near one of its line foci, sees the spokes parallel to that line black and the spokes at right angles grey. The angle of the black spokes is the axis of the astigmatism, and adding cylindrical power until every spoke looks equally black measures its size. Young’s observation, that horizontal and vertical lines were sharp at different distances, is the same chart with only two spokes.

The curve for the circle of least confusion is flat: at that focus every orientation is blurred alike. An astigmatic eye that accommodates to the circle sees every line equally grey, which is why mild astigmatism is often noticed not as distortion but as eyestrain, the eye hunting between its two foci and settling for the compromise in which nothing is sharp.

Why a sphere cannot fix it #

A spectacle lens with ordinary spherical surfaces adds the same power in every direction. Put one in front of an astigmatic eye and it moves both line foci by the same amount — closer or further — without changing the distance between them. It can choose where between the two lines the retina sits, and the best it can do is put the circle of least confusion there.

For an eye with a 4-millimetre pupil and a dioptre and a half of astigmatism, the best sphere leaves 10.3 arcminutes of blur — ten times the eye’s resolution, enough to make text a line or two smaller than normal unreadable. A cylindrical lens, which has power in one meridian and none in the other, does something no sphere can: it changes one focal length without the other, and set at the right axis with the right power it moves one line focus onto the other. The two lines become one point, and the eye then behaves as if it had a single focal length, which an ordinary sphere can put on the retina. Airy’s cylinder did exactly that for his left eye, and the cylinder and axis in a modern prescription are the same correction.

Most eyes have some astigmatism, mostly because the cornea is curved slightly more steeply in one direction than the other — it is a toric surface, as the ground lens was. Half a dioptre is common and usually left uncorrected, because its circle of least confusion is within the eye’s own depth of focus. A dioptre and a half is corrected almost always.

Finding the axis by asking #

Measuring an eye’s astigmatism means finding two numbers, a power and an angle, by asking a person which of two blurs looks better — and the asking is organised by the geometry of the conoid. The standard tool, introduced by Edward Jackson in 1887, is a cross-cylinder: a lens with a small positive power in one meridian and an equal negative power in the meridian at right angles, so that its average power is zero and it adds pure astigmatism. Held in front of an eye already given its best sphere, and flipped between its two positions, it lengthens one of the eye’s line foci and shortens the other, or the reverse.

If its axes are lined up with the eye’s own, one flip makes the eye’s astigmatism larger and the other smaller, and the patient sees one position as clearly worse; the cylinder in the trial frame is then increased or reduced until the two flips look equally bad, which happens when the eye’s astigmatism has been cancelled and the cross-cylinder’s own is all that is left. If its axes are set at 45° to the eye’s, the flips are symmetric about the eye’s astigmatism in a different way, and the patient’s preference says which way to rotate the trial cylinder’s axis. The procedure converges in a minute or two on the power and axis to a quarter of a dioptre and a few degrees, and at no point does the patient need to describe the blur — only to compare two. It works because the circle of least confusion, held on the retina by the best sphere, responds symmetrically to equal and opposite astigmatism added along the right axes, and asymmetrically otherwise.

A mirror turned to the side #

A lens does not need to be ground toric to be astigmatic. Any perfectly spherical lens or mirror becomes astigmatic as soon as light reaches it at an angle. A spherical mirror of focal length $f$, lit by a parallel beam arriving at an angle $i$ to its axis, has two foci, worked out by Henry Coddington in 1829: in the plane containing the beam and the axis, at $f\cos i$, and across that plane, at $f/\cos i$.

Seen obliquely, the mirror’s curvature looks different in the two directions: foreshortened in the plane of incidence, unchanged across it. A mirror tilted by a few degrees has line foci millimetres apart, which at the apertures of real instruments is far more than their depth of focus. That is one reason a telescope mirror is pointed along its axis and its images degrade towards the edge of the field. It is why a spectrograph that folds its light with spherical mirrors arranges for the astigmatic focus that is a line along the slit to fall on the detector — each point of the slit then becomes a short line along the slit’s own length, which costs nothing in resolving the spectrum. And it is why the folded cavities of many lasers contain an element cut at an angle: the folding mirrors of a ring or a Z-shaped cavity, which the two mirrors that hold a beam found must keep a beam inside them over thousands of passes, make it astigmatic at every bounce, and a crystal or plate set at Brewster’s angle in the beam is astigmatic in the opposite sense and cancels it.

Astigmatism at the edge of every photograph #

Even a perfectly made spherical lens shows the same effect for objects off its axis, which reach it at an angle. It is the third of the five classical departures from perfect imaging that the surface that images one point exactly listed, and the one whose blur is a pair of lines at different distances rather than a patch. Each point at the edge of the field is imaged as two short lines, one pointing towards the centre of the picture and one at right angles to that, at slightly different distances behind the lens. The flat scene that comes back curved found that a lens images a flat scene onto a curved surface, and astigmatism splits that surface into two — a tangential surface, where the radial lines are sharp, and a sagittal one, where the circles round the centre are — with the Petzval surface lying between them.

A camera lens whose design has brought those two surfaces together, so that off-axis points are imaged as points again, is called an anastigmat, and the word was a selling point when the first such lenses appeared in the 1890s. The design problem is the same one the spectacle cylinder solves, in a harder form: the astigmatism varies across the field, and it has to be cancelled everywhere at once by the choice of the lens’s curvatures, glasses, spacings and, above all, the position of its stop. The condition a lens must meet followed the earlier step in the same campaign, the removal of coma; astigmatism and field curvature were the next two to fall.

Astigmatism put to work #

Astigmatism is usually a defect to be removed, but the conoid’s peculiarity — that the shape of the spot, and not just its size, depends on the focus — makes it a sensor. Put a weak cylindrical lens in front of a detector divided into four quadrants. When the spot is exactly at the circle of least confusion it is round and lights the four quadrants equally. When the focus moves one way the spot becomes an ellipse along one diagonal, and the other way along the other diagonal. The difference between the diagonal pairs of quadrants is a signal that says not only how far out of focus the system is but in which direction — something an ordinary blurred spot, which grows the same way on either side of focus, cannot say. That is how the pickup in a compact-disc or DVD player has kept its laser focused on a spinning disc to within a fraction of a micrometre: an astigmatic focus-error detector driving the lens up and down hundreds of times a second.

The same idea locates single molecules in three dimensions. In localisation microscopy, each fluorescent molecule appears as a spot whose centre can be found to a few nanometres in the plane of the image; its depth is invisible in an ordinary microscope, since the spot blurs symmetrically above and below focus. A weak cylinder in the imaging path makes the spot elliptical along one direction above focus and along the other below it, and the ellipse’s shape gives the depth to a few tens of nanometres. Since 2008 this has produced three-dimensional images of structures inside cells far finer than the diffraction limit, with an aberration doing the measuring. The plane of focus a tilted lens tilts found another case in which a lens deliberately set out of its ideal arrangement does something the ideal one cannot; astigmatism is the oldest.

What the figures leave out #

The figures trace rays by the paraxial rule for each meridian separately and treat the lens as thin, which is exact for the astigmatism itself but leaves out the spherical aberration, coma and diffraction that a real astigmatic lens also has. Diffraction matters most: the line foci are not infinitely thin lines but have a width set by the aperture, as the hole that makes the sharpest picture found for any aperture, and an astigmatism smaller than the diffraction blur is invisible. The eye’s figures take the blur to be geometrical and ignore its other aberrations, the pupil’s changing size, and the brain’s adaptation to a long-standing astigmatism, which makes newly corrected astigmatism look strange for a few days. Coddington’s equations are drawn for a thin pencil of rays; a wide beam at a large angle has coma as well. The domain is an astigmatism large compared with diffraction and small enough for the paraxial treatment of each meridian to hold.

Still open: how much astigmatism an eye should have #

Infants are commonly born with substantial astigmatism, which mostly fades in the first years of life as the cornea and lens grow — the same process, emmetropisation, that brings most eyes to a focal length matching their length. How the growing eye senses its own astigmatism and corrects it, and whether a blur that is different in different directions steers the growth of the eye’s shape, are active questions, studied in animals reared with cylindrical lenses and in children’s eyes followed over years. The answers matter for myopia, which is rising steeply in many countries and whose development seems linked to how the eye’s growth responds to blur. Why some astigmatism persists in so many adult eyes — whether it is a residue the growth process tolerates or something it is aiming for — is not known.

The optics is exact. A lens with focal lengths $f_v$ and $f_h$ in two meridians brings a parallel beam to two perpendicular lines, at $f_v$ and $f_h$, and to a circle of least confusion at their harmonic mean, $2f_vf_h/(f_v + f_h)$ — 111.1 mm for 100 and 125 — whose radius is $(f_h - f_v)/(f_h + f_v)$ of the aperture’s; no sphere can do better than that circle, 10.3 arcminutes of blur for a 1.5-dioptre eye at a 4 mm pupil, and a cylinder, by moving one line onto the other, removes it. A focus is a point only for a surface that is the same in every direction, and an eye that is not is corrected by a lens that is not either.