Last updated: 2026-07-13

Lens power and the diopter

The optical power of a thin lens measures how strongly it bends light. It is defined as the reciprocal of the focal length:

When the focal length $f$ is in metres, the power $P$ comes out in diopters (symbol D), where $1\ \text{D} = 1\ \text{m}^{-1}$. A lens that focuses parallel light at $0.5\ \text{m}$ has a power of $1/0.5 = 2\ \text{D}$; halve the focal length to $0.25\ \text{m}$ and the power doubles to $4\ \text{D}$. Shorter focal length means stronger bending and higher power.

Sign convention

Power inherits the sign of the focal length. A converging (convex) lens brings parallel rays to a real focal point and is given a positive focal length, so its power is positive. A diverging (concave) lens spreads parallel rays apart; they only appear to come from a virtual focal point on the incoming side, which is assigned a negative focal length, so the power is negative. A $+3\ \text{D}$ lens and a $-3\ \text{D}$ lens bend light equally hard but in opposite senses.

Formula

Because power is a reciprocal length, it turns the awkward reciprocal arithmetic of focal lengths into simple addition — the main reason opticians and physicists quote lenses in diopters.

Worked example

A magnifying glass forms a sharp image of the sun at a focal length of $20\ \text{cm}$. Converting to metres and taking the reciprocal:

Working backwards, a lens rated at $2.5\ \text{D}$ has a focal length of $f = 1/2.5 = 0.4\ \text{m}$, or $40\ \text{cm}$.

Combining two lenses

Place two thin lenses so they touch, and their powers add directly:

A $+2\ \text{D}$ lens against a $+3\ \text{D}$ lens behaves like a single $+5\ \text{D}$ lens, with an effective focal length of $1/5 = 0.2\ \text{m}$. Adding a diverging element subtracts: a $+5\ \text{D}$ lens combined with a $-2\ \text{D}$ lens yields $+3\ \text{D}$. If the two powers cancel — say $+4\ \text{D}$ and $-4\ \text{D}$ — the pair has zero net power and behaves like a flat window, with the focal point pushed to infinity.

Where lens power is used

Optical power in diopters is the natural language of any system built from stacked lenses: microscope and telescope objectives, camera lenses, and the compound lenses inside projectors. Because contacting powers add, designers can reason about a multi-element system by summing simple numbers rather than combining focal lengths. The diopter also makes it easy to compare lenses at a glance — a larger magnitude always means stronger bending.

Limitations

The relation $P = 1/f$ describes an ideal thin lens, where the thickness is negligible compared with the focal length. The additive rule $P_\text{total} = P_1 + P_2$ assumes the lenses are thin and in contact; separating them by a distance $d$ introduces a correction term, $P_\text{total} = P_1 + P_2 - d\,P_1 P_2$. Thick lenses, wide ray angles, and different colours of light (which refract by slightly different amounts) all shift the effective power away from the simple reciprocal, and a full optical design accounts for these effects.