Lensmaker's Equation Calculator

Background.

A lens works because light slows down inside glass, and a curved face meets rays at different heights at different angles, so each ray is bent by a different amount. Shape the faces correctly and a beam of parallel light converges to a single point. The distance from the lens to that point, the focal length, is the number an optician grinding spectacles or a student building a simple telescope actually needs.

The lensmaker's equation predicts it from two things: what the lens is made of and how its faces are curved. The material enters as n − 1, where n is the refractive index. Glass with an index of 1.5 carries light at two-thirds of its speed in a vacuum, and the 0.5 left after subtracting 1 measures how strongly the glass bends light compared with the air around it. The shape enters as the curvatures 1/R1 and 1/R2 of the front and back faces: tight curves with small radii bend light hard, gentle curves only a little, and a flat face contributes nothing.

Signs carry the shape. In the Cartesian convention used here, light travels from left to right and a radius is positive when that surface's centre of curvature lies to the right of it, beyond the surface. A biconvex lens therefore has a positive first radius and a negative second one, and a biconcave lens the reverse. Get a sign wrong and a converging lens turns into a diverging one in the answer.

The lens is treated as thin compared with its radii and as sitting in air. One index is used for all light, so the spread of focus by colour is left out, along with aberrations, aspheric surfaces and the behaviour of multi-element systems.

What is lensmaker's equation calculator?

The focal length of a lens is the distance from the lens to the point where rays travelling parallel to its axis come together, or, for a diverging lens, the point they appear to spread from. It is positive for converging lenses and negative for diverging ones. Its reciprocal, in inverse metres, is the lens's power in dioptres, the unit printed on eyeglass prescriptions. The lensmaker's equation predicts it from the lens material and the curvature of its two faces.

How to use this calculator.

1. 01Enter the refractive index of the lens material; it must be greater than 1. Typical values are about 1.49–1.50 for acrylic and CR-39 spectacle plastic, about 1.517 for BK7 crown glass and about 1.59 for polycarbonate.
2. 02Enter the first radius, for the face the light reaches first, in metres: positive if that face bulges out toward the incoming light, negative if it is hollowed toward it.
3. 03Enter the second radius by the same rule. The back face of a biconvex lens bulges away from the light, so its centre of curvature sits on the incoming side and its radius is negative.
4. 04For a flat face, enter a very large radius such as 1000000; zero is refused. Read the signed focal length — positive converges, negative diverges — and take 1 ÷ f for the power in dioptres.

The formula.

1/f = (n − 1)(1/R1 − 1/R2). Here n is the refractive index of the lens relative to the surrounding air, R1 and R2 are the signed radii of the first and second surfaces in metres, and f is the focal length in metres. Each surface has a power of (change in index) ÷ R. Light enters the first face going from air into glass, giving (n − 1)/R1, and leaves the second going from glass into air, giving (1 − n)/R2; in the thin-lens approximation the two simply add, which is where the minus sign comes from. The radii dominate: halve both and the power doubles, halving f. The material acts through n − 1, so raising n from 1.5 to 1.6 lifts that factor from 0.5 to 0.6 and shortens f by a sixth. An index of 1 or less is refused, as is any pair of radii whose curvature terms cancel, because the focal length would be infinite.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, University Physics Volume 3, section 2.4, Thin Lenses.
2. [2]Hecht, Optics, 5th ed. (PRINT).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.