From the shape of a lens to its focal length
The focal length of a lens depends on two things: how strongly its faces are curved and how much the material slows light down. The lensmaker’s equation links them. Opticians and lens designers use it to choose the radii that give a required power.
The formula
For a lens of thickness d in a medium:
1/f = (n − 1) × [1/R₁ − 1/R₂ + (n − 1) d ÷ (n R₁ R₂)]
- n is the refractive index of the lens divided by that of the medium around it (1.0 for air).
- R₁ and R₂ are the radii of curvature of the first and second faces.
- d is the thickness at the center.
For a thin lens, d is small and the last term disappears: 1/f = (n − 1)(1/R₁ − 1/R₂).
The optical power in diopters is the medium index divided by the focal length in meters. Eyeglass prescriptions are written in diopters.
Sign convention
With light traveling from left to right, a radius is positive if its center of curvature lies to the right of the surface and negative if it lies to the left. A flat face has an infinite radius; enter 0 and the calculator treats it as flat. A positive focal length means a converging lens, a negative one a diverging lens.
Worked example
A symmetric biconvex lens of index 1.5, with R₁ = +100 mm, R₂ = −100 mm and a center thickness of 5 mm, in air:
- Thin-lens focal length: 1 ÷ [0.5 × (0.01 + 0.01)] = 100 mm.
- With the thickness term, the focal length becomes 100.84 mm.
- Power: 1 ÷ 0.10084 m = 9.92 diopters.
For most eyeglass and magnifier lenses, the thin-lens value is close enough. Thick lenses, such as ball lenses or microscope objectives, need the full formula.
Refractive index matters
The same shape in a higher-index material bends light more. Crown glass is about 1.52, polycarbonate 1.59 and high-index eyeglass plastics up to 1.74. That is why a strong prescription in high-index plastic is thinner and lighter.
The surrounding medium matters too. Put the example lens in water (index 1.33) and the relative index drops to about 1.13, so the focal length grows to roughly 400 mm, four times longer than in air. This is why your vision blurs under water and why a diving mask, which keeps a layer of air in front of the eyes, restores sharp sight.
The equation assumes rays close to the axis. Real lenses also suffer aberrations, which designers correct by combining several elements.