The physics of the eye

One converging system, one fixed screen

Light entering the eye is bent twice. Most of the bending happens at the cornea, the curved front window, because that is where light crosses the largest change of refractive index. The lens behind it makes the smaller adjustment, and the iris opens and closes the pupil to control how much light gets in. For calculations the whole arrangement is treated as one thin converging lens, with the retina acting as the screen about 23 mm behind it.

So the image on the retina is real, inverted and diminished. Nothing later turns it the right way up. The brain simply learns to read it, which is why the standard question asks about the image and not about what you experience.

Three equations from the lenses lesson do all the arithmetic here. Power and focal length are two names for one property,

with f in metres and P in dioptres. Object distance, image distance and focal length are tied together by

in the real is positive convention, and the size of the image follows from

Accommodation: the power that changes

Here is what the opening claim gets wrong. The retina cannot move, so v is stuck at about 23 mm whatever you look at. Read 1/u + 1/v = 1/f with v fixed and one thing has to give. When u changes, f must change with it, and the eye must supply a different power for every distance. Ciliary muscles do it by squeezing the lens into a fatter, more strongly converging shape. That is accommodation.

Two distances mark the limits of what accommodation can reach. The far point is the furthest object the eye can focus, at infinity for a healthy eye, and the eye is fully relaxed there. The near point is the closest, taken as 0.25 m for a normal adult eye and reached with the lens working hardest. That 0.25 m is also called the least distance of distinct vision, and it is the distance every reading calculation assumes.

Two defects, two lenses

Myopia, short sight, is the eye that is too powerful for its own length, so parallel rays from a distant object converge in front of the retina and arrive at it already spreading again. Distant objects blur. The eye's far point is no longer at infinity but at some finite distance, and near vision is usually fine.

Hypermetropia, long sight, is the opposite. The eye has too little power for its length, so rays from a close object would meet behind the retina. Close objects blur, and the near point has moved further out than 0.25 m.

A correcting lens is not asked to form an image on the retina. It is asked to move the object to somewhere the eye can already cope with, and the image it makes is the object the eye then works on. That image is virtual, on the same side as the object, so it carries a negative v.

Astigmatism is a different fault. The cornea, or sometimes the lens, is not the same shape in every direction across its face, so it has more power along one axis than along the one at right angles to it. A point object cannot be brought to a point image in both directions at once, and the patient sees lines sharp in one orientation and blurred in the other.

The cure is a lens carrying matching extra power along one axis only, so a prescription needs three numbers rather than one. SPH is the ordinary spherical power in dioptres. CYL is the extra cylindrical power, again in dioptres. AXIS is an angle in degrees from 0 to 180 saying which direction the cylinder acts along. A prescription reading −2.50 / −0.75 × 90 is a myopic eye whose astigmatism is corrected on the vertical axis.

The retina as a photodetector

As a detector the eye is answering two different questions at once, and it has two kinds of cell to answer them. Cones work in bright light, come in three types responding to different bands of wavelength, and between them give colour vision. Rods work in dim light, come in one type only, and so report brightness and nothing about colour.

That figure is the eye's spectral response, and it is the reason the visible spectrum ends where it does. Sensitivity peaks in the yellow-green, near 555 nm in good light, and falls to nothing outside roughly 400 nm to 700 nm. The rods peak at a shorter wavelength, near 500 nm, so as the light fades the eye's best colour shifts towards blue and reds go dark first.

Spatial resolution is decided by how the cells are wired, not by how many there are. Cones crowd into the fovea, a small patch on the axis of the eye, and each one there has effectively its own nerve fibre. Two lit cones with an unlit one between them therefore arrive at the brain as two separate signals, and detail survives. Rods lie outside the fovea and many of them share a single fibre, so their signals are added together before they leave the eye.

That sharing is a bargain, not a fault. Adding the light caught by many rods makes a signal large enough to notice when almost no light is arriving, which is why the dark-adapted eye can work by starlight. What it costs is detail, because the brain cannot tell which rod in the group was lit. Cones make the opposite trade, and the eye's practical resolution of about one minute of arc, roughly 3 × 10⁻⁴ rad, is a fovea figure. Look slightly to one side of a faint star and it brightens, because you have moved its image off the fovea and onto the rods.