The lens maker’s equation tells optical designers how to shape glass to get a desired focal length. Convex surfaces facing the light have positive radius; a symmetric biconvex lens has R₁ = −R₂.
How to use the Lens Maker’s Equation Calculator
- Radius of curvature R₁
- Radius of curvature R₂
- Lens thickness d
- Refractive index n
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter both radii with sign (positive when the centre of curvature is beyond the surface), the centre thickness and the index of the glass. Use a very large radius for a flat surface.
Inputs explained
| Field | What to enter |
|---|---|
| Radius of curvature R₁ | Units: mm, cm, m, in, µm |
| Radius of curvature R₂ | Units: mm, cm, m, in, µm |
| Lens thickness d | Units: mm, cm, m, in, µm |
| Refractive index n | A number |
What each result means
- Focal length f = 1÷(([Refractive index n] – 1)×(1÷[Radius of curvature R₁] – 1÷[Radius of curvature R₂] + ([Refractive index n] – 1)×[Lens thickness d]÷([Refractive index n]×[Radius of curvature R₁]×[Radius of curvature R₂])))
- Optical power = 1÷[Focal length f]
Example values
How the main result changes when Radius of curvature R₁ changes and everything else stays at the example values above:
| Radius of curvature R₁ (mm) | Focal length f (mm) |
|---|---|
| 50 | 65.2404 |
| 75 | 83.7431 |
| 100 | 97.5804 |
| 125 | 108.3193 |
| 150 | 116.8956 |
| 200 | 129.7357 |
Formula
1/f = (n − 1)·[1/R₁ − 1/R₂ + (n − 1)·d / (n·R₁·R₂)]
Worked example
Inputs — Radius of curvature R₁: 100 mm; Radius of curvature R₂: -100 mm; Lens thickness d: 5 mm; Refractive index n: 1.5168.
Results — Focal length f = 97.58041 mm; Optical power = 10.248 D.
Key terms
- Radius of curvature
- Radius of the sphere a lens surface is part of.
Frequently asked questions
What is the sign convention?
R is positive if the centre of curvature is on the outgoing side of the surface.
What does a negative focal length mean?
A diverging lens.
Can I ignore thickness?
For thin lenses yes; the last term then vanishes.
How does n affect f?
Higher index glass gives a shorter focal length for the same shape.