Lensmaker's Equation Calculator
Find a lens's focal length and power in diopters from its refractive index and the radii of curvature of both surfaces with the lensmaker's equation — thin or thick lens, in air or any medium.
How it works
1/f = (n_rel − 1) [ 1/R₁ − 1/R₂ + (n_rel − 1) d ÷ (n_rel R₁ R₂) ] · n_rel = n_lens ÷ n_medium · thin lens (d = 0): 1/f = (n_rel − 1)(1/R₁ − 1/R₂) · power P = n_medium ÷ f (in metres) diopters
The lensmaker's equation links a lens's focal length to its shape and material. It uses the Cartesian sign convention with light travelling left to right: a radius is positive when the surface's centre of curvature lies to the right of the surface. So a biconvex lens has R₁ > 0 and R₂ < 0, and a biconcave lens has R₁ < 0 and R₂ > 0. A flat face has an infinite radius (enter 0). Immersing a lens in a medium closer to its own index, such as water, makes it much weaker.
Worked example
A thin (d = 0) biconvex glass lens with n = 1.5 in air, R₁ = +100 mm and R₂ = −100 mm: 1/f = (1.5 − 1)(1/100 − 1/(−100)) = 0.5 × 0.02 = 0.01 per mm, so f = 100.00 mm. Its power is 1000 ÷ 100 = 10.000 D and it is a converging lens.
Sources: HyperPhysics — Lensmaker's formula · Wikipedia — Lens (optics): lensmaker’s equation
Frequently asked questions
What is the lensmaker's equation?
1/f = (n − 1)(1/R₁ − 1/R₂) for a thin lens in air, where n is the lens’s refractive index and R₁, R₂ are the radii of its two surfaces. The thick-lens version adds the term (n − 1)d/(nR₁R₂).
What sign should the radii have?
With light coming from the left, a surface whose centre of curvature is on the right (beyond it) has a positive radius. For a biconvex lens R₁ is positive and R₂ negative; for a plano-convex lens with the curved side first, R₁ is positive and R₂ is flat (0 here).
What happens to a lens in water?
The relative index falls from 1.5 to 1.5/1.333 = 1.125, so (n_rel − 1) drops from 0.5 to 0.125 and the focal length becomes about four times longer.
How do I convert focal length to diopters?
In air, power in diopters = 1 ÷ focal length in metres. A 100 mm lens is 10 D and a −250 mm lens is −4 D.
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This calculator is for educational purposes. Double-check important results before acting on them.