Formula and Method for Diopter (Lens Power)
A diopter (symbol D, unit m⁻¹) is the standard unit of optical power for lenses, mirrors, and the eye. It is defined as the reciprocal of the focal length measured in meters: D = 1/f. A short focal length means a strongly bending, high-power lens (large D); a long focal length means a weak lens (small D). This calculator converts a focal length in any common unit into diopters, identifies whether the lens is converging or diverging, and — if you enter a second lens — adds the two powers together to find the combined power and combined focal length of the pair.
How the calculation works
First, the focal length is converted to meters (1 in = 0.0254 m, 1 cm = 0.01 m, 1 mm = 0.001 m). The power in diopters is then the reciprocal of that value: D = 1/f(m). For example, a lens with f = 50 cm = 0.5 m has a power of D = 1/0.5 = 2.00 D. If you supply a second lens's focal length, the calculator converts it the same way and computes its power D2, then adds the two powers directly — D_total = D1 + D2 — because this is the standard thin-lens-in-contact approximation used throughout optics and optometry. The combined focal length is simply f_total = 1/D_total.
Sign convention and common mistakes
- Converging vs. diverging: by the standard sign convention, a converging (convex) lens has a positive focal length and therefore positive power (+D); a diverging (concave) lens has a negative focal length and negative power (−D). Enter a negative value for the focal length of a diverging lens.
- Don't average focal lengths: when stacking two lenses, never average or add the raw focal lengths — powers add, focal lengths do not. Two +2.00 D lenses in contact give +4.00 D total (f = 25 cm), not a focal length that is the sum of the individual focal lengths.
- Lens separation matters: the simple addition D_total = D1 + D2 assumes the lenses are thin and touching. For lenses separated by a distance d (in meters), the more general formula is D_total = D1 + D2 − d·D1·D2.
Real-world applications
- Eyeglass and contact lens prescriptions are written directly in diopters (e.g., −3.00 D for myopia, +1.50 D for hyperopia or a reading add).
- Stacking corrective lenses — such as reading glasses worn over prescription lenses, or a magnifying loupe over safety glasses — uses D_total = D1 + D2 to predict the effective power of the combination.
- Camera and projector optics specify lens power in diopters when working with close-up (macro) attachment lenses, which add their power to the main lens.
- The relaxed human eye has a total optical power of roughly 60 D, produced mainly by the cornea (~40 D) and the crystalline lens (~20 D) acting together.