Optics

The eye that loses its lens in water

Eyes opened under water see only a blur, and the reason is not water in the eye or salt or murk. The human eye does two-thirds of its focusing not with its lens but with the curved front of its cornea, whose power comes from the jump in refractive index between air and tissue. Water is almost as dense optically as the cornea, and the jump all but disappears: about forty-three of the eye's sixty dioptres vanish, and the focus moves to three times the eye's own length behind the retina. A mask gives the cornea back its air; a small pupil and a hard squint buy back a little; animals that hunt in both worlds have flattened their corneas and given the work to the lens.

Assumes: What a lens is doing, and why three rays are enough · The angle past which light cannot leave

What a lens is doing explains a lens by its surfaces. A curved boundary between two transparent materials bends light by an amount set by its curvature and by the difference between the materials’ refractive indices, and a lens is two such boundaries. The power of a single surface, in dioptres, is the step in refractive index across it divided by its radius of curvature in metres. Nothing in that sentence mentions glass, or the lens as an object; it mentions a step and a curve.

The human eye is usually drawn with its lens at the centre, as the part that focuses. That is half true at best. The lens is the part that changes its focus, by being squeezed or relaxed by the muscles around it, and it carries about a third of the eye’s power. The other two-thirds are in the front surface of the cornea, the clear dome at the front of the eye, because that is where the largest step in refractive index lies: from the air, at 1.000, to the corneal tissue, at 1.376. Change what is in front of the cornea and the eye’s focus changes with it. Fill the space in front of it with water and most of the eye’s focusing disappears.

Where the eye’s focusing is done

The arithmetic can be done with a simplified model of the eye, following the schematic eyes that Allvar Gullstrand built for his Nobel-winning studies of the eye’s optics.

The cornea’s front surface has a radius of about 7.8 millimetres. In air the step across it is from 1.000 to 1.376, a difference of 0.376, and divided by the radius that is a power of 48 dioptres. Its back surface, curving the other way into the aqueous humour at 1.336, has a small negative power of about six dioptres. The lens, relaxed, adds about nineteen. In all, the eye has about sixty dioptres of power, enough to bring parallel light from a distant object to a focus on the retina twenty-two millimetres behind its principal plane.

In water, at 1.333, the step at the front of the cornea falls to 0.043. Divided by the same radius, it gives five and a half dioptres. Forty-three dioptres have gone. The back of the cornea and the lens are inside the eye, bathed on both sides by the eye’s own fluids, and do not notice the water at all. The eye goes from sixty dioptres to eighteen.

The focus that moves out of the eye

Losing forty-three dioptres does not make the eye a little short-sighted or long-sighted. It moves the focus out of the eye altogether.

The model eye’s eighteen dioptres under water would focus parallel light seventy-three millimetres behind its principal plane, three times as far back as the retina. On the retina itself the rays from each point of a distant scene have converged only a third of the way, and they land spread over a disc almost as large as the pupil. Every point is a blur of the same size, and the scene is a uniform smear of its own colours. An ordinary long-sighted eye is out by one to five dioptres, and corrected with spectacles of that power; the eye under water would need a spectacle lens of forty-three dioptres, an extraordinarily strong lens, mounted in water.

This is why the blur seems total and why it is not helped by blinking or squinting in the usual way. The ordinary range of the eye’s own adjustment, its accommodation, is about ten dioptres in a child and two or three in a middle-aged adult. It can push the focus only in one direction — adding power, for near objects — and here it would need to add forty. There is no object distance at which an unaided human eye under water can focus at all.

How much the pupil can buy back

What the eye cannot fix with power it can partly fix with aperture. The blur disc into which a point is spread is, in angle, the pupil’s diameter times the remaining defocus.

A smaller pupil means a narrower bundle of rays from each point, which converges less and spreads less on the retina even though its focus is in the wrong place. This is the principle of the hole that makes the sharpest picture: a pinhole camera has no lens at all and still forms an image, whose sharpness is set by the size of the hole. The eye’s pupil can narrow to about two millimetres in bright light, and a child’s eye can accommodate by ten or twelve dioptres. Together they cut the blur of a relaxed eye with a three-millimetre pupil in half.

In 2003 Anna Gislén and colleagues at Lund University tested the underwater sight of children of the Moken, a sea-nomad people of the Andaman coast of Thailand and Myanmar who gather shellfish from the sea floor. The Moken children resolved gratings under water about twice as fine as European children tested in the same way, and photographs of their eyes showed why: under water their pupils narrowed to under two millimetres, smaller than in bright air, and they accommodated as hard as their eyes allowed. European children, given some weeks of training, learned to do the same. The improvement is real and large in proportion, and it stops far short of normal sight, because no pupil small enough to remove a forty-dioptre blur would let in enough light to see by.

What a mask does to the view

The full answer is to put the air back. A diving mask traps a pocket of air in front of the eyes behind a flat glass window, and the cornea, facing air again, recovers its power.

The mask’s window is flat, and a flat boundary between water and air refracts too. Light coming from an object in the water bends away from the normal as it passes into the air in the mask, and so reaches the eye at a larger angle than it left the object. Near the centre of the view every angle is multiplied by about four-thirds, the ratio of the refractive indices. Objects look a third larger and, by the same factor, a quarter nearer than they are, a familiar experience for divers, who learn to reach a little further than they think. Further off-axis the stretching grows. A ray arriving in the water at 48.6 degrees from straight ahead leaves the window at ninety degrees, grazing along it: the whole hemisphere of view in the air of the mask is filled by a cone of water only ninety-seven degrees across.

That cone is the same one a fish sees the sky through. The angle past which light cannot leave derives the critical angle for light going from water to air, 48.6 degrees, and a fish looking up sees the whole sky compressed into a circle of that half-angle overhead, Snell’s window, with the reflected underwater scene outside it. A diver in a mask is in the same situation turned round: looking out from air into water, with a whole hemisphere of the air side mapped onto the critical cone of the water side. A mask’s field of view is narrowed for the same reason the fish’s sky is small.

The flatter the cornea, the less it loses

Animals that need to see both in air and in water have had to solve the same problem, and the physics leaves only a few solutions open.

The loss is the change in index step, 0.333, divided by the cornea’s radius. A strongly curved cornea, like the human one, loses a great deal; a flat cornea loses nothing, because it had no power to lose. An eye with a flat cornea must do all of its focusing with its lens, which is unaffected by the outside medium, and then sees equally well in air and in water. Several diving birds and seabirds have corneas much flatter than land birds’, and penguins’ are close to flat; their lenses are correspondingly more powerful.

Fish, which live entirely in water, have given up the cornea as a focusing surface altogether. Their corneas have almost no power, and their lenses are spheres of graded refractive index, densest at the centre, that do all the work: the lens with no axis follows the gradient that lets such a lens form a sharp image despite its spherical shape, a solution worked out by James Clerk Maxwell and found in fish eyes. Fish taken out of water are short-sighted for the opposite reason: their small residual corneal power jumps up in air. Some animals adjust actively. Diving birds such as cormorants squeeze their soft lenses into the opening of the iris and accommodate by tens of dioptres, far beyond any human eye, to restore the lost power under water.

Why an air eye puts its power at the front

The human arrangement is not an accident; it is the cheapest way to focus light that arrives through air. A single curved surface between air and water-like tissue has an index step nearly ten times larger than any step available inside the eye, where all the materials have indices between 1.33 and 1.41. Putting most of the power at that first surface lets the lens be thin and soft, which is what allows it to change shape for accommodation, and keeps the eye short. A fish eye, with no such step available, needs a lens of the highest index biological tissue can reach, densest at its core, and a lens that is a sphere: it cannot accommodate by changing shape and instead moves the whole lens back and forth, like a camera.

The front-heavy design has costs in air too. A strongly curved single surface brings rays at the edge of the pupil to a focus slightly in front of those near the centre — spherical aberration — and the eye corrects it partly by flattening the cornea towards its edge and partly by a lens whose index falls from centre to rim. It also has a curved field: a single strong surface images a flat scene onto a curved surface, the effect the flat scene that comes back curved traces in camera lenses, and the eye answers by curving its retina to match, which no camera sensor can easily do. And its power varies with colour, because tissue disperses light: the eye is about two dioptres stronger for blue than for red, the uncorrectable fringe of the colour fringe no aperture can close, which the brain tolerates and partly ignores.

Depth of focus, the other face of the blur

The pupil’s narrowing helps under water by trading light for depth of focus, and it is worth seeing that this is the same trade the eye makes in bright air. The focus that is a slab, not a plane shows that a lens does not focus sharply at one distance but over a range, whose thickness grows as the aperture shrinks: the blur of an object slightly out of focus is the aperture times the defocus, so halving the aperture doubles the range of distances that look sharp. A two-millimetre pupil in bright sunlight gives an eye a depth of focus of about half a dioptre either side of where it is focused, which is why the ageing eye, whose accommodation is failing, reads more easily in bright light than in dim.

Under water the same arithmetic applies to a defocus of forty dioptres instead of half a dioptre. The pupil cannot narrow enough to bring the blur down to normal acuity: to reach one minute of arc against thirty dioptres of residual defocus it would need to be a hundredth of a millimetre across, a pinhole that would let in too little light to see by and would itself blur the image by diffraction. The narrowing pupil buys back a factor of two or so, which is the whole of the Moken advantage, and the rest must come from the mask.

The same step at every boundary

The eye’s problem is a particular case of a general principle: the refracting power of any curved surface depends on what is on both sides of it. A glass lens of a camera, focused in air, is badly out of focus under water and must be housed behind a port; a flat port magnifies by four-thirds and distorts the field, as the mask does, and underwater camera housings often use a dome port, a hemispherical window centred on the lens, whose curvature has the opposite effect and preserves the field at the cost of a virtual image close to the dome. Contact lenses work because their front surface takes over the role of the cornea’s; under water they are as useless as the cornea itself.

The problem of a boundary that does most of the work and then disappears has an acoustic twin. The lever that lets air speak to water explains the bones of the middle ear as an impedance transformer that couples sound in air to the fluid of the inner ear, and under water that transformer is mismatched in the opposite direction: sound arrives in the water and reaches the inner ear partly through the bones of the skull, bypassing the eardrum, and the ear loses its ability to tell where the sound came from. A human head is designed for air in both senses, and its sharpest instruments are at the boundary with the air.

How sharp is sharp enough under water

A diver’s view through a mask is limited by other things than the eye. Water absorbs red light within a few metres and scatters blue and green, and particles in the water scatter light as the atmosphere does in fog, reducing contrast with distance. Visibility in clear tropical water may be thirty metres and in a temperate harbour less than one. How far apart two things have to be gives the eye’s resolution, about one minute of arc in air, set by the pupil’s diffraction and the spacing of the retina’s cones; behind a mask that resolution is restored in principle, and in practice is limited by the water’s own blurring of the scene long before the eye’s limit is reached.

An eye of round numbers

The model eye uses round numbers for the radii and indices of the cornea and treats each surface as a thin refractor with the powers simply added, which is adequate for a comparison between air and water but not for computing the eye’s focus precisely, since real eyes are thick optical systems with the separation of their surfaces mattering. The lens is given a fixed power of nineteen dioptres; a real lens has a graded index and changes shape with accommodation. The blur estimate is geometric, pupil times defocus, and ignores diffraction, which is negligible at these defocuses, and the eye’s aberrations, which are not. The mask figure treats its window as a single flat boundary and ignores the glass’s thickness, which shifts the image slightly without changing the angles. The radii for flattened corneas are illustrative; the measured corneal curvatures of diving species vary widely.

Still open: how much the brain adds

The Moken children’s advantage under water is optical — a small pupil and strong accommodation — but the training study that taught European children the same trick found the improvement took weeks, which suggests that learning to constrict the pupil and accommodate together in an unusual context is a skill rather than a reflex. How far the brain also learns to interpret a blurred underwater image better, by sharpening its processing of low-contrast edges as it does in people with long-standing refractive error, has not been separated from the optical gain. Whether the advantage lasts into adulthood, when accommodation declines, is also open: Moken adults were reported to be less adept than their children, as the optics of an ageing lens would predict.

The habit worth carrying away is to ask where the step is. The eye’s sharpest focusing surface is not its lens but the front of its cornea, whose 48 dioptres come from the step in index from air to tissue; under water the step falls from 0.376 to 0.043 and 43 dioptres vanish, leaving the focus 73 mm behind the retina — so a mask restores sight by restoring the air, and a flat cornea, which has no step to lose, sees as well under water as above it. A lens is a boundary, and a boundary depends on what is on the other side.