Diopter Calculator
Enter a focal length to get the optical power in diopters, or enter diopters to get the focal length. Switch to the thin-lens mode to solve for image distance and magnification from object distance, or use the combined-lens tab to find the effective power of two lenses separated by a known gap. All calculations update instantly in your browser.
Formula
Worked example
A converging lens with a 20 cm focal length: f = 0.20 m, so P = 1 / 0.20 = 5 D. Place an object 40 cm from the lens: 1/di = 1/0.20 - 1/0.40 = 5 - 2.5 = 2.5, so di = 0.40 m = 40 cm. Magnification m = -40/40 = -1 (real, inverted, same size as object).
What is a diopter?
A diopter (symbol D) is the SI unit of optical power, defined as the reciprocal of the focal length measured in metres. A lens that focuses parallel light to a point 1 metre away has a power of 1 D; a lens focusing light to 25 cm has 4 D. Converging (convex) lenses have positive diopter values because they cause light rays to bend toward a real focal point. Diverging (concave) lenses have negative diopter values because they cause rays to spread, forming only a virtual focal point on the same side as the incoming light. Diopters appear most famously on eyeglass and contact-lens prescriptions, where a minus sign means the person is myopic (nearsighted) and a plus sign means hyperopic (farsighted).
The thin-lens equation
For a single thin lens the relationship between focal length (f), object distance (do), and image distance (di) is 1/f = 1/do + 1/di, or equivalently 1/di = P - 1/do where P is the lens power in diopters and distances are in metres. A positive di means the image is on the opposite side from the object (a real image, which can be projected onto a screen). A negative di means the image is on the same side as the object (a virtual image, seen by looking back through the lens, as with a magnifying glass). The lateral magnification m = -di/do: a negative magnification means the image is inverted relative to the object, and |m| > 1 means it is enlarged. If the object is placed exactly at the focal point, the rays emerge parallel and the image forms at infinity.
Combined lens systems and vergence
When two thin lenses are placed in contact the combined power is simply the sum P_total = P1 + P2. When they are separated by a distance d (in metres), the effective power is P_total = P1 + P2 - d * P1 * P2. This formula underlies the design of telescope and camera zoom systems. Vergence is an alternative way to track the same physics: it measures the curvature of the wavefront at any given plane, in diopters. Diverging light arriving at a lens carries a negative vergence equal to -1/do; after passing through a lens of power P the output vergence is U_out = U_in + P. A positive output vergence means the light is converging and will form a real image; a negative output vergence means it is still diverging and will form a virtual image.
Diopters in eyeglass prescriptions
An eyeglass prescription lists sphere (Sph), cylinder (Cyl), and axis values, each in diopters and fractions of 0.25 D. The sphere value corrects the main focusing error: negative for myopia, positive for hyperopia. The cylinder and axis correct astigmatism, where the cornea or lens has different curvatures along different meridians. A prescription of -2.00 Sph means the eyeglass lens has a power of -2 D, which pushes the focal point back to fall exactly on the retina instead of in front of it. Presbyopia, the loss of near-focus flexibility with age, is corrected by reading-glass lenses in the range of +1 to +3 D.
Typical optical powers by application
| Optical element / situation | Typical power (D) | Focal length |
|---|---|---|
| Human cornea | ~43 | ~2.3 cm |
| Human crystalline lens (relaxed) | ~17 to 25 | ~4 to 6 cm |
| Total human eye | ~60 | ~1.7 cm |
| Reading glasses (+) | +1 to +3 | 33 to 100 cm |
| Nearsighted correction (-) | -0.5 to -10 | -10 to -200 cm |
| Farsighted correction (+) | +0.5 to +8 | 12.5 to 200 cm |
| Standard magnifying glass | +4 to +10 | 10 to 25 cm |
| Camera telephoto (500 mm) | ~2 | 50 cm |
| Camera wide-angle (18 mm) | ~55.6 | 1.8 cm |
Approximate diopter values for common lenses and optical structures. Individual values vary significantly.
Frequently asked questions
What does a diopter measure?
A diopter measures the optical power of a lens or mirror: how strongly it bends light. Mathematically it equals one divided by the focal length in metres. A 5 D lens focuses parallel light to a point 20 cm away. Higher diopter values mean a stronger, more curved lens.
How do I convert focal length to diopters?
Divide 1 by the focal length expressed in metres. For example, a 50 cm lens: f = 0.50 m, P = 1 / 0.50 = 2 D. If you have the focal length in centimetres, divide 100 by it to get diopters directly: P = 100 / f_cm.
What is the difference between a positive and a negative diopter?
A positive diopter (plus sign) means the lens converges light to a real focus. These are convex lenses used for farsightedness, magnifiers, and camera lenses that form real images. A negative diopter (minus sign) means the lens diverges light; only a virtual focus forms. These are concave lenses used for myopia correction.
How powerful is the human eye in diopters?
The total optical power of the relaxed human eye is roughly 60 D, made up of about 43 D from the cornea and about 17 to 25 D from the crystalline lens. The eye can add up to 15 D more by changing the shape of the lens (accommodation), allowing focus at different distances.
What is the thin-lens equation and when does it apply?
The thin-lens equation is 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance, all in metres. It applies to any single lens that is thin compared to its focal length. It predicts whether the image is real or virtual, its distance from the lens, and its magnification. For thick lenses or multi-element systems you need the general matrix (ABCD) formulation.
How does vergence relate to diopters?
Vergence is the power of a wavefront, measured in diopters. It equals 1 divided by the distance (in metres) at which the light would converge or diverge. Passing through a lens adds the lens power to the incoming vergence: U_out = U_in + P. Vergence analysis lets you track a wavefront step by step through any number of optical surfaces without re-solving the lens equation from scratch each time.