About the Lensmaker's Equation Calculator
This lensmaker's equation calculator finds the focal length of a lens from the way it is made: the radius of curvature of each surface, the refractive index of the lens material, and (optionally) the centre thickness and the refractive index of the medium around it. It returns the focal length, the optical power in dioptres, whether the lens converges or diverges light, and the back focal length measured from the rear surface.
It is useful for physics and optics students, amateur telescope makers grinding their own lenses, and anyone comparing how the same lens shape would behave in different glasses or plastics — a bar chart shows the focal length you would get from eight common materials with the radii you entered.
Radii follow the standard Cartesian sign convention: a surface is positive when its centre of curvature lies on the far (outgoing) side of the lens, so a symmetric biconvex lens has R₁ positive and R₂ negative. Enter 0 for a flat surface. With thickness 0 the thin-lens form is used; a positive thickness switches to the full thick-lens equation. Results are paraxial (rays close to the axis) and ignore aberrations.
With the default inputs, the focal length f (in your length unit) is 20. Change any value above to recalculate instantly.
How to use the lensmaker's equation calculator
- 1Enter the refractive index of the lens material.
- 2Enter both surface radii using the sign convention (biconvex: R₁ positive, R₂ negative; 0 for flat).
- 3Add the centre thickness for a thick lens, or leave it at 0.
- 4Pick the length unit and, if the lens is not in air, the medium’s index.
- 5Read the focal length, power and lens type, and compare materials in the chart.
Formula and method
The lensmaker's equation links a lens's focal length to its shape and material. Each curved surface refracts light by an amount set by its curvature (1/R) and by how much the lens index differs from the surrounding medium. For a thin lens the two surfaces simply add; for a thick lens the extra term accounts for light travelling through the glass between them. A flat surface has infinite radius, so its 1/R term is zero.
The focal length here is the effective focal length, measured from the lens’s principal plane; the back focal length, f(1 − (nᵣ − 1)d/(nᵣR₁)), is the distance from the rear surface to the focus, which is what you actually measure on a bench. Optical power in dioptres is the medium index divided by the focal length in metres. A positive focal length means a converging lens and a negative one a diverging lens. The equation is paraxial and ignores aberrations and the slight variation of refractive index with wavelength.
- f
- Effective focal length
- n
- Refractive index of the lens material
- nₘ
- Refractive index of the surrounding medium (air = 1)
- nᵣ
- Relative index n/nₘ
- R₁, R₂
- Radii of curvature of the first and second surfaces (signed)
- d
- Centre thickness of the lens
- P
- Optical power in dioptres (1/m)
Worked examples
Symmetric biconvex glass lens
1/f = (1.5 − 1)(1/20 − 1/(−20)) = 0.5 × 0.1 = 0.05 per cm, so f = 20 cm. In metres that is 0.2 m, a power of 5 dioptres.
Plano-convex crown glass lens
The flat back surface contributes nothing, so 1/f = 0.517 × (1/10) = 0.0517 per cm and f = 19.34 cm (5.17 D).
Biconcave lens
1/f = 0.5 × (−1/15 − 1/15) = −0.0667 per cm, so f = −15 cm: a diverging lens with −6.67 D of power.
Thick biconvex lens, 10 mm thick
1/f = 0.5 × [1/50 + 1/50 + 0.5 × 10 / (1.5 × 50 × (−50))] = 0.5 × (0.04 − 0.001333) = 0.019333 per mm, so f = 51.72 mm, slightly longer than the 50 mm thin-lens value. The back focal length is 51.72 × (1 − 0.5 × 10 / (1.5 × 50)) = 48.28 mm.
The same glass lens under water
The relative index drops to 1.5 / 1.333 = 1.1253, so 1/f = 0.1253 × 0.1 and f ≈ 79.8 cm — four times weaker than in air.
Frequently asked questions
What is the lensmaker's equation?+
For a thin lens in air, 1/f = (n − 1)(1/R₁ − 1/R₂), where n is the refractive index and R₁ and R₂ are the signed radii of the two surfaces. It predicts the focal length of a lens from how it is ground.
What sign convention is used for the radii?+
With light travelling left to right, a radius is positive if its centre of curvature lies to the right of the surface. A biconvex lens therefore has R₁ > 0 and R₂ < 0; a biconcave lens has R₁ < 0 and R₂ > 0.
How do I enter a flat surface?+
A flat (plano) surface has an infinite radius of curvature, so its 1/R term is zero. Enter 0 for that radius and the calculator treats it as flat, as for a plano-convex or plano-concave lens.
Why is a lens weaker under water?+
Refraction depends on the difference between the lens index and the medium index. Glass (1.5) in water (1.333) has a relative index of only about 1.125, so the bending is roughly a quarter of what it is in air.
What is the difference between focal length and back focal length?+
The effective focal length is measured from the principal plane inside the lens; the back focal length is measured from the rear glass surface to the focus. For thin lenses they are almost equal, but they differ noticeably for thick lenses.
How do I convert focal length to dioptres?+
Optical power in dioptres is 1 divided by the focal length in metres (for a lens in air). A 20 cm lens has a power of 1/0.2 = +5 D; a −50 cm lens has −2 D.