The eye a fly cannot afford

The beam that does not spread and the price it pays showed how a narrow aperture spreads a wave, and how the attempts to avoid this always involve paying somewhere else. How far apart two things have to be found the resulting limit for a lens, which is simple: the smallest angle a telescope or an eye can resolve is about $\lambda/D$, and to do better needs a larger aperture. A person with good sight can read small print across a room because a pupil a couple of millimetres across resolves about an arcminute.

An insect has a head a millimetre or two across, and a very different eye. Instead of one aperture and one retina it has thousands of small apertures, packed on a curved surface, each collecting light from one narrow direction and each feeding its own small group of receptors. This is a compound eye, and each unit is an ommatidium. It is an evolutionary success story, having arisen more than 500 million years ago, and has been used by more species than any other kind of eye. What is not at all obvious is why insects do not simply use more facets, which would seem to give them a sharper picture of the world. A short calculation shows they cannot, and it gives a surprising answer to what a sharp compound eye would have to be.

Two blurs, and a width between them #

The argument is an old one, and Horace Barlow set it out in 1952 with a clarity that has not been improved on. Imagine a spherical eye of radius $R$, whose surface is divided into facets of width $D$. The directions in which adjacent facets look differ by an angle

$$\Delta\phi = \frac{D}{R}.$$

No detail smaller than about this angle can be represented, since the eye’s sampling grid has that spacing. Thus smaller facets, packed more closely, are better.

Each facet, though, is a small aperture, and light passing through a hole of width $D$ spreads by an angle

$$\Delta\rho = \frac{\lambda}{D}.$$

The dark lines a wave cannot avoid gave this for a single slit. So smaller facets are worse here. These two effects oppose each other, and there has to be a best width.

Adding the two blurs in quadrature gives

$$\Delta^2 = \left(\frac{D}{R}\right)^2 + \left(\frac{\lambda}{D}\right)^2.$$

Differentiating with respect to $D$ and setting the result to zero, the minimum occurs where the two contributions are equal, at

$$D = \sqrt{\lambda R}, \qquad \Delta_{\min} = \sqrt{\frac{2\lambda}{R}}.$$

For a 1 mm eye in green light, $D$ is about 22 µm, and the smallest possible blur is about 1.8 degrees. That is not a matter of a bad design, and it would hold for an eye made with perfect optics, because the best such eye is limited by the head’s size. Even with an ideal design, the fine detail a fly can see is fixed, at about a hundred times worse than a person’s.

It is worth comparing the two limits precisely. For a lens eye, the blur falls as the inverse of size. For a compound eye, it falls as the inverse square root. The whole gulf between the two designs comes from the difference between those exponents. Barlow’s main contribution was his empirical check: he collected the values from 27 species of Hymenoptera, and found facet widths close to $\sqrt{\lambda R}$ in those with different head sizes. A large bee and a small wasp have facets of different sizes for reasons that follow from diffraction.

The rule the facets roughly follow #

The insects shown are typical examples, for orientation and not for detailed analysis. Published values can vary with the eye region measured. Still, they are consistent with the line within reasonable tolerance. There are no eyes that lie far below it, since a facet that narrow gains nothing and loses light. Many eyes lie somewhat above it, for a reason that has nothing to do with resolution. A small facet admits little light. A photoreceptor counts photons, and at low light levels the shot noise in that count limits what it can detect: to tell one grey from another by a difference of 1 per cent requires about ten thousand photons. Insects active at dusk have facets wider than the diffraction rule requires, which improves their sensitivity at the expense of sharpness. Allan Snyder made that trade-off precise in the 1970s, showing that the best design depends on the brightness of the scene.

It is common to find variation across the same eye. A dragonfly, which hunts insects on the wing, has large facets looking forward and upward, with an interommatidial angle under half a degree, and much smaller ones elsewhere. A male fly chasing a female has an enlarged “love spot” in the part of the eye that looks at her. In all those cases the local radius of curvature changes as well: in the acute zone the surface is flatter, so that a larger number of facets look into a smaller solid angle, and it behaves as though it were part of a much larger sphere. This is what the formula says should work: only the local radius matters, so an eye can be locally sharp at the cost of being larger or having worse vision elsewhere. A similar concentration of resources happens in a human fovea, where acuity is about one arcminute and falls away by a factor of ten within ten degrees or so.

What human sharpness would cost #

The formula can be turned around. To resolve an angle $\Delta$ with a compound eye, the radius has to be

$$R = \frac{2\lambda}{\Delta^2}.$$

At one arcminute, which is about the human limit, $R$ is about 12 m. To achieve the same result with a lens takes an aperture of $\lambda/\Delta$, or 1.7 mm. The compound eye’s radius rises as the square of the sharpness desired. An eye twice as sharp is four times as large, in radius. Kurt Kirschfeld, in 1976, published a well-known cartoon of a person with a compound eye sufficient to give human acuity. If such an eye only had to be sharp in the forward direction, like a human fovea, its diameter would be reduced to about a metre; even then, it would look absurd on a human frame. Taken over the entire field at one arcminute, the figure is far larger.

It is a pretty result, since it says something that no study of insect anatomy would suggest. The barrier between the two kinds of eye is a matter of how an aperture is used. Each facet’s diffraction limit is set by its own small width, whereas a lens collects all of its light through the whole pupil and maps each direction to a separate point. Its aperture is available to all directions at once. A compound eye has divided that same area into thousands of independent apertures and thereby discarded the phase relation between them that would have given a sharp image. That relation is exactly what a telescope array has to preserve, and the fringe that measures a star shows the great gain that comes with doing it.

So why have compound eyes at all? Apart from history, the design has real advantages at small sizes. It needs no focusing mechanism, and its depth of field is effectively infinite. It covers a very wide field, around 300 degrees or more, without any eye movement, and each facet can respond very fast. A housefly’s receptors can follow flicker up to about 250 Hz, against about 60 for a human, and neural processing near the receptors is arranged accordingly. For a small flying animal doing fast aerial manoeuvres at short range, these are good trades. An eye with a lens needs a long enough focal length for a good retina, and on a very small head, this is hard to accommodate. Jumping spiders, however, have managed the alternative at an impressively small size, using an elongated tube and a lens.

An eye made of straws #

In a real ommatidium there is a small lens at the surface, called a corneal facet, which focuses light on to the tip of a light-guiding rod, called a rhabdom. The rhabdom acts as a waveguide about a micrometre in diameter, with the same physics as the cone a fibre will accept. Its width is close to that of a wavelength. Thus it supports only a few modes, and those are what Snyder analysed: an optical fibre of this size has a restricted set of angles at which light can be accepted, adding its own contribution to the blur. A further small effect is due to the eye having screening pigment that blocks light from entering one facet and being absorbed in the next. The pigment sits between facets, as a kind of wall, in the manner of the straws in the drawing.

The picture shows what these two limits look like geometrically. On the left, the cones are narrow, because a wide aperture makes a sharp beam, but there are gaps in coverage, or at least large steps in the sampling. On the right, there is no shortage of facets, but each facet’s cone overlaps several neighbours, and they look at nearly the same part of the world. An ideal design has its cones just touching. Nyquist’s criterion from the speeds a pulsed radar folds into one sets the spacing that a sampling lattice needs for a given pattern, about two samples per cycle, so a more careful analysis gives a constant slightly different from the 1 used here. Those details change the coefficient, not the square root. It is a simple but robust fact that the eye’s sampling and its optics are tuned to each other, in many species.

A human retina shows the same kind of match. Its cones in the fovea are spaced about 2.5 µm apart. With a focal length of about 17 mm, that gives about 30 arcseconds between samples, and so a Nyquist limit of about one arcminute. The diffraction limit for a 3 mm pupil is roughly the same, about one arcminute. Thus nature has matched the receptor grid to the optical blur there as well, since a finer grid gains nothing and a coarser grid wastes optics. The eye that loses its lens in water looked at the other end of that same optical system.

A pixel count fixed by the head #

Dividing a hemisphere’s area, $2\pi R^2$, by a facet’s area, $D^2 = \lambda R$, gives a striking formula:

$$N = \frac{2\pi R}{\lambda}.$$

The number of facets goes as the radius, not as the area. That is because a larger eye uses larger facets. For a 1 mm eye, this gives about 12,600 facets on a hemisphere. Real eyes usually have fewer, and a housefly’s has about 3,000 for each eye, since it spends some of its head’s capacity on other things and its eye’s surface is not a full hemisphere. A dragonfly has up to about 30,000, with eyes of larger radius. Compared with this, a human eye has about 120 million rods and 6 million cones in an eye of 12 mm radius, and the fovea alone gives far better resolution than any compound eye of comparable size could have.

$R/\lambda$ here is the only combination that appears. Interestingly, this is also the number that decides whether the optical calculations in terms of rays or waves apply. The information capacity of any aperture is related to it, as with the invariant that is a count, where the number of independent modes that can pass through an optical system was found to be a fixed count. An eye of radius $R$ with a hemispherical view and one small aperture per direction uses only about $R/\lambda$ of its possible modes. A lens eye of the same radius might use about $(R/\lambda)^2$, or thousands of times more. An important part of what evolution discarded, in the design of the compound eye, is the great majority of the information that the head’s aperture area could have carried.

This is also the reason for some attempts to improve artificial cameras with insect-inspired designs. Modern compound lens systems on phones use several small apertures with overlapping views, and computational methods recover some of what the small apertures lose, by exploiting known overlaps between them. They preserve the angular coverage and small depth of a compound eye, and yet they pay the same scaling cost for sharpness. A phone with four identical small cameras can match, in resolution, one with a single lens only up to a limited factor, and physics provides the reason.

Why a lens works on a head that small, and when it does not #

There is a further limit on a lens eye that a compound eye escapes. A lens eye needs its focal length to reach a retina with fine enough receptors, and those receptors cannot be much narrower than about a micrometre, since a thinner waveguide no longer confines light of that wavelength. That sets a minimum focal length for a given acuity. At very small sizes, then, a lens eye runs out of room, and a compound eye is a good way to use a small space. Very small wasps of the family Mymaridae, barely 200 µm long, have eyes with only tens of facets, and at that size there is hardly anything better to do. The hole that makes the sharpest picture found a similar square-root law for the best pinhole, where a similar balance arises between geometry and diffraction: the sharpest pinhole’s diameter also goes as the square root of a length and a wavelength. The two results come from the same two terms.

This is a useful and unexpected connection. A pinhole camera’s blur has a geometric part proportional to the hole’s size, and a diffraction part that is inversely proportional to it, and the best hole diameter is about $\sqrt{\lambda f}$, with $f$ the distance to the screen. Exchanging $f$ for $R$ produces Barlow’s rule. An ommatidium is, in effect, a pinhole camera with one pixel, and the eye is thousands of those side by side on a sphere. Nautilus has an actual pinhole eye and has always been regarded as an evolutionary puzzle, since a lens would have been so much better.

What the estimate leaves out #

The simple formula uses a slit’s diffraction, whereas a real facet is a small hexagonal lens whose diffraction pattern is closer to an Airy disc and whose light passes into a waveguide; a more careful treatment changes the constant a little. The quadrature sum is an approximation to the actual convolution of two blur functions. The spacing term also requires some care, for sampling theory says that a sharp point is what counts at the Nyquist limit and not simply the angle between facets. All of those affect the constant and not the square root. Published refinements typically give a best facet slightly different from $\sqrt{\lambda R}$, by amounts less than a factor of 1.5.

An insect’s actual acuity also depends on the brain and not only on its optics. Motion can be used by some insects to see better than the static sampling limit would suggest. A hoverfly moves its receptors slightly in a way that may enhance its acuity, a process also found in fruit flies, which has been called photomechanical. The figure for insect values should be read as typical, and no attempt was made to represent the range across species. The essay takes the eye to be a sphere with a uniform pattern, and this is only sometimes close to the truth. The comparison at one arcminute also assumes bright light, at the human daytime limit.

Still open: how much a moving eye can beat its own sampling #

The diffraction limit is firm, but the sampling term can be evaded if the receptors move. An eye that scans a scene samples each direction more than once. Recent measurements on fruit flies by Mikko Juusola and colleagues in 2017 found that a photoreceptor contracts quickly in response to light, moving and narrowing its field of view in a way that could allow the animal to see detail finer than its facet spacing would allow, which they called hyperacuity. In 2024, their work reported that binocular overlap adds a stereo sense at the same fine scale. How much of that actually reaches the behaviour of a real insect is under active investigation, since a measured neural signal does not always correspond to what the animal uses.

The same questions have started to appear in bio-inspired sensors. Artificial compound eyes with thousands of facets have been made on curved substrates, and they show wide fields and good motion sensitivity. Whether those can be combined with computation to give more than a compound eye’s normal resolution remains an open engineering problem. The basic scaling, with its square root, sets the ceiling, and designers are looking at how far movement can be used to bring the real system nearer to it.

A facet of width $D$ on an eye of radius $R$ blurs by $D/R$ from its spacing and $\lambda/D$ from its diffraction, and the two balance at $D = \sqrt{\lambda R}$, for a blur of $\sqrt{2\lambda/R}$. A 1 mm head cannot beat about 1.8°. A blur that falls only as the inverse square root of size means that human sharpness costs a compound eye twelve metres of radius and a single lens 1.7 mm of pupil. The difference is in using one aperture or thousands.