Lens Power in Diopters

Optical power in dioptres is just the reciprocal of the focal length in metres, and the reason opticians prefer it to focal length is arithmetic convenience of a very practical kind. Focal lengths of thin lenses in contact combine as $1/f = 1/f_1 + 1/f_2$, which nobody wants to do at a fitting bench. Powers simply add: a $+2.0$ D reader over a $+0.5$ D correction is $+2.5$ D, and you have done it in your head. The whole of clinical refraction is built on that single property, and the unit is named for it — a dioptre is a reciprocal metre wearing a clinical hat.

A $-4.0$ D prescription for myopia is a diverging lens with $f = -0.25$ m, or $-25$ cm. A $+2.5$ D reader has $f = +40$ cm, which is roughly where you would hold a book. The relaxed human eye is about a $+60$ D system altogether, and the split is instructive: roughly $+43$ D comes from the cornea, where air meets tear film, and only about $+19$ D from the crystalline lens the muscles can adjust. That is why refractive surgery reshapes the cornea — it is where nearly three quarters of the power lives. Accommodation adds up to about $+14$ D of extra power in a child and declines steadily to nearly nothing by the sixties, which is presbyopia expressed as one number.

The reciprocal-metre form pays off a second time in what optometrists call vergence. The reciprocal of a distance in metres is the vergence of the light arriving from or heading to that point, so the thin-lens equation itself becomes plain addition in dioptres: incoming vergence plus lens power equals outgoing vergence. Light from an object 50 cm away arrives with $-2$ D of vergence; add a $+6$ D lens and it leaves with $+4$ D, focusing 25 cm beyond. No reciprocals, no common denominators. This is how the arithmetic is actually done in a clinic, and it is worth knowing even if you never sit in that chair.

Four traps. Metres, always metres. A 50 mm camera lens is $+20$ D, not $+0.02$ D, and entering millimetres puts the answer out by a factor of 1000 — the commonest error on this page by a wide margin. Second, the sign carries the whole clinical meaning: converging positive, diverging negative, so a myopia correction is negative and a hyperopia or reading correction is positive. Dropping a minus sign converts one condition into its opposite. Third, the additive rule requires the lenses to be genuinely in contact, or separated by a distance negligible compared with their focal lengths. Separate them by $d$ and the combination becomes $P = P_1 + P_2 - d P_1 P_2$, which is exactly why a spectacle lens sitting 12 mm in front of the eye is not the same prescription as a contact lens on the cornea — above about $\pm 4$ D the vertex-distance correction is clinically significant and is routinely calculated. Fourth, the cylinder and axis figures on a prescription describe a separate astigmatic correction in a particular meridian, and this single spherical number says nothing about them.