Lens maker's formula

Advanced

Focal length from curvature and index.

Key relation1/f = (n−1)(1/R₁−1/R₂)

1/f = (n − 1)(1/R₁ − 1/R₂) gives focal length from refractive index and curvatures.

Worked example

Biconvex lens with R₁ = 20 cm, R₂ = −20 cm, n = 1.5: f = 20 cm.

Common mistake

Placed in a medium, replace n by n_lens/n_medium.

In more detail

The lensmaker's formula 1/f = (n − 1)(1/R1 − 1/R2) gives the focal length of a lens from its refractive index and surface radii.

Opticians and lens manufacturers design lenses using the lensmaker's formula.
Students use it to link lens shape to focal length.
Lens makers choose the curvature of each surface and the glass index to get the focal length they need.

What do R1 and R2 mean?

The radii of curvature of the two surfaces, with sign conventions for convex and concave.

Why does refractive index matter?

Higher refractive index bends light more strongly, shortening the focal length for the same shape.

Why does the glass index matter?

A higher index bends light more strongly, so a lens with the same curvature has a shorter focal length.