Result

Thin lens equation

\frac{1}{d _{o}} + \frac{1}{d _{i}} = \frac{1}{f}, \frac{1}{f} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})

Statement

Relates object distance, image distance, and focal length; the lensmaker's equation gives f from the lens's shape and refractive index.

Symbols and units

$d_{o}, d_{i}$	object, image distance	m
$f$	focal length	m
$n$	refractive index of the lens	—
$R_{1}, R_{2}$	radii of curvature of the two surfaces	m

When it applies

A thin lens (thickness negligible compared with the radii of curvature and object/image distances), paraxial rays.

Don't use when

Assumes a thin lens (negligible thickness compared with radii of curvature and object/image distances) and paraxial rays — a thick lens needs a more general treatment.

Derivation

Takes exactly the same form as the mirror equation; the lensmaker's equation follows from applying refraction at each of the lens's two curved surfaces in turn.

\frac{1}{f} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})

Watch out for

A converging (convex) lens has positive f; a diverging (concave) lens has negative f. The thin lens and mirror equations share an identical form — once one is known, so is the other, with a different sign convention for which side is 'real'.

Connects to

Mirror equation