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 where the thickness along the optical axis (drawn as a horizontal blue line) is small compared to its focal length, the situation looks as shown in fig. 1.

lens equation

Fig. 1.

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 or film) 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)

Finally, 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. It is a surprisingly powerful tool to calculate e.g. magnification or depth of field of a lens.