Lens Equation

A converging lens is characterized by collecting parallel light rays in a focal point F. As a consequence, light emitted from a point in an object plane (parallel to the lens, at a finite distance) is collected in a single point in a corresponding image plane (also parallel to the lens). For a thin converging lens, its thickness along the optical axis (drawn as a horizontal blue line) is small compared to its focal length.

Basic geometry of the thin lens equation.

The lens equation describes the relation between focal length f of the lens, object distance g between object and lens and image distance h between lens and image sensor when the image of the object appears sharp. It can be derived as follows.

By similar triangles along the purple ray passing through F left of the lens, we have

G / (g – f) = H / f (L1)

Likewise, by similar triangles along the purple ray through F right of the lens, we get

G / f = H / (h – f) (L2)

Solving both equations for G / H results in

(g – f) / f = f / (h – f) (L3)

After some simple algebraic transformations, the equation looks like this

g h = f h + f g (L4)

After division by f g h, we get the well-known form

1 / f = 1 / g + 1 / h (L5)

This equation is known as the lens equation or thin lens formula. See how g and h are adjusted when the lens is focused in the next section.