Inside the Eye: How We See

Light is bent (refracted) where it passes between materials with different refractive index n. Four surfaces do the work: the front and back of the cornea and the front and back of the lens. A picture of the world is drawn, smaller and upside down, on the retina. The brain turns it back.

A curved surface of radius R between indices n1 and n2 has power P = (n2 - n1)/R.

Together, with the 3.6 mm gap: about 58.9 D; Gullstrand's classic eye has 58.6 D. The cornea does about two thirds of the bending, the lens a third. A plain glass-like lens (n = 1.386) would give only 13.3 D: the real lens has a graded index, so this page uses an effective index (model value).

To see something close, the ciliary muscle squeezes, the zonule fibres go slack and the lens becomes rounder and stronger. The power added is A. An object at 25 cm needs A = 1/0.25 = 4 D. A 10-year-old can add about 15 D, a 60-year-old almost none (the average amplitude is 18.5 - 0.3 × age; people differ by 2-3 D).

In the model the lens surfaces get steeper, the lens thicker and its effective index higher as A grows (model values). The slider is calibrated so that A is exactly the shift of the far point.

If the eyeball is too long, light from far away focuses in front of the retina. Each millimetre of extra length adds roughly -2.6 D of error; the model gives -2.59 D per mm for +1 mm, -4.9 D for +2 mm and +2.8 D for -1 mm (far-sighted). The farthest sharp point (far point) is at 1/|F| metres: 38.7 cm for -2.6 D.

Glasses sit about 12 mm in front of the cornea. A lens of power F_s in the glasses plane gives the cornea the effect F; a short-sighted eye needs a slightly stronger lens in glasses than at the cornea:

Check: the far point is 38.7 cm from the cornea = 37.5 cm from the glasses, and 1/0.375 = 2.67 D. (The first version of the spec had the sign of this correction wrong; it is fixed and tested below.)

A defocus of ΔD dioptres through a pupil of diameter p makes a blur disc of angle β ≈ p × ΔD. For p = 4 mm and 1 D that is 4 milliradians = 13.8 arcmin, while the eye can normally tell apart details of 1 arcmin. The Eye-cam blurs the picture by the size of the spot the traced rays make on the retina, combined with the diffraction limit (0.77 arcmin for a 3 mm pupil).

Diffraction through a 3 mm pupil blurs a point to 1.22 λ/D = 0.77 arcmin. One arcmin on the retina is 4.86 µm; the cones in the fovea are 2-3 µm apart. The eye is a very good camera. (Text only in this version.)

• All surfaces are smooth, rotationally symmetric conics; rays are drawn in one plane (meridional), no astigmatism.
• The graded-index lens is replaced by an equivalent lens whose index grows with accommodation; the conic constants of the lens are tuned so that spherical aberration is almost zero for a 4 mm pupil. Real eyes keep small aberrations.
• No colour (chromatic) error, the pupil is fixed at 4 mm, no scattering.
• Real eyes differ from person to person: treat every number as a range, not a point. Values: Gullstrand schematic eye; Hofstetter amplitude; they should be checked against the primary sources before publication.
• The cornea and lens are drawn as semi-transparent surfaces, not as full light-transmitting glass.