Optical Power Calculator (Diopters)

Optical power describes how strongly a lens bends light. It is measured in diopters (D), which are simply inverse meters.

Power = 1 / focal length (in meters)

A lens with a 0.5 m focal length has 2 diopters of power. A lens with a 25 cm focal length has 4 diopters.

Sign convention:

• Converging (convex) lens: positive focal length, positive power
• Diverging (concave) lens: negative focal length, negative power

Eye prescriptions use diopters directly. A −2.0 D prescription means a diverging lens correcting myopia (nearsightedness). A +1.5 D means a converging lens correcting hyperopia (farsightedness).

Thin lens equation:

1/f = 1/do + 1/di

Where do = object distance (positive if on the incoming light side), di = image distance (positive if on the outgoing side, meaning real image; negative = virtual image).

Magnification:

m = −di / do

Negative m means the image is inverted. |m| > 1 means the image is larger than the object.

Lenses in contact: when two thin lenses touch, the total power is just the sum:

P_total = P₁ + P₂

An optometrist combining a sphere and cylinder correction exploits this directly.

Give it any two of the three and it finds the third

Focal length, object distance, image distance. Enter focal length alone and you get the power in diopters. Enter any two and the thin lens equation fills in the missing one, along with the magnification and whether the image is real or virtual.

The most common version of this problem is focal length plus object distance: you know the lens and you know where the subject is, and you want to know where to put the film, the sensor, or the screen. Rearranged, that is

di = f · do / (do − f)

Watch what happens as do approaches f. The denominator goes to zero and di runs away to infinity: an object sitting exactly at the focal point produces no image at all, because the rays leave the lens parallel. That is not a failure of the equation, it is how a collimator works, and it is why a projector cannot focus on a subject placed at its own focal length.

Real versus virtual, and why the sign matters

A positive image distance means the light actually converges there. Put a piece of paper at that spot and a picture appears on it. That is a real image, and it is always inverted for a single converging lens.

A negative image distance means the rays only appear to come from that point. Nothing lands on paper held there. That is a virtual image, upright, and on the same side of the lens as the object. Every magnifying glass held closer to the page than its focal length produces one, which is exactly why you can see the enlarged letters but cannot project them onto a wall. A diverging lens produces a virtual image at every object distance, no exceptions.

A quick reality check on prescriptions

Reading glasses sold as “+2.00” have a focal length of 50 cm. That is the whole meaning of the number. A −4.00 D myopia correction is a diverging lens with a 25 cm focal length, and the minus sign is doing real work: it tells you the lens spreads light rather than gathering it. If a prescription and a focal length ever disagree in sign, one of them has been copied down wrong.