Defects of Vision & Spectacles

Many of us wear lenses in front of our eyes every day. Each pair of spectacles is a small calculation: the lens that moves the world to where your eye can focus it.

The eye as a lens with a fixed screen#

The retina is a fixed distance v₀ (about 2.5 cm in an adult) behind the eye-lens. To focus objects at different distances u₀, the eye changes its focal length. In magnitudes:

• The far point is reached with the muscles relaxed (largest f). In a normal eye f_max = v₀, so the far point is at infinity.
• The near point is reached with the muscles fully strained (smallest f); about 25 cm or less for a normal eye.

Nearsightedness (myopia)#

A nearsighted eye sees close objects but not distant ones. Even fully relaxed, its focal length is too short (the lens is too thick, or the eyeball too long), so parallel rays focus in front of the retina. Straining only makes it worse.

The cure is to make the rays slightly divergent before they enter: a diverging lens. If the far point is at distance x, the lens must form a virtual image of a distant object at x, so its focal length is −x:

Farsightedness (hypermetropia)#

A farsighted eye sees distant objects but its near point is well beyond 25 cm, so reading is uncomfortable. Even fully strained, its focal length isn’t short enough (the lens is too thin, or the eyeball too short). Rays from 25 cm would focus behind the retina.

A converging lens makes them less divergent. If the near point is at y, the lens must form an image of an object at 25 cm at distance y. With u = −25 cm and v = −y:

Presbyopia

With age the ciliary muscles weaken and can’t strain enough, so the near point recedes even in an otherwise normal eye. This farsightedness from age is presbyopia, corrected the same way.

Astigmatism#

If the eye’s surfaces are curved differently in different planes, lines in some directions look sharp while lines at right angles look blurred. A quick test: draw several lines through a point, equally dark. If one looks darkest and the line at right angles faintest, and this swaps when you turn the page by 90°, the eye is astigmatic. The cure is a cylindrical lens, curved differently in different planes to compensate.