# Lens maker equation

1/f = (n-1)*(1/R_1-1/R_2)
Lens maker equation.
Enter 'x' in the field to be calculated.
if flat (R = infinity) enter 1e10 m

Diagram: Example of a converging meniscus lens composed of two spherical diopters of radius R_1 and R_2.

The lens maker equation computes the focal length of a lens, from the radii of curvature of the two spherical diopters that compose it (face 1 and 2 in the diagram) and n, the refractive index of the lens material. The optical power of the lens can be easily deduced using the formula f = 1 / V (See Calculate the optical power of a lens.)

When projected on the plane, the lens has two sides (see diagram above):
- Face 1: circle of center C_1 and radius R_1
- Face 2: circle of center C_2 and radius R_2

It is assumed that the lens is thin ie its thickness e is small relative to the radii of curvature R_1 and R_2 (e << R_1 and e << R_2). The lens focal length is given by the lens maker formula :

1/f = (n-1)*(1/R_1-1/R_2)

R_1: Radius of curvature of the first surface.
R_2: Radius of curvature of the second surface.
n: Lens material refractive index
f: Lens focal length

If one of the faces of the lens is plane then the radius is infinity (R = oo). The formula remains valid, just omit the corresponding term: 1/R = 0. In the calculator, simply enter a large number like 1e10 m as radius (= 10^10 m).

## Sign convention

The radii of curvature R_1 and R_2 may be positive or negative depending on the shape of the surfaces of the diopters constituting the lens.

Directional distances :
R_1 = \bar{S_1C_1}
R_2 = \bar{S_2C_2}

So,
- R_1 > 0 if S_1 is convex
- R_1 < 0 if S_1 is concave
- R_2 > 0 if S_2 is concave
- R_2 < 0 if S_2 is convex

Lentille Type R_1 R_2
()
biconvex R_1 > 0 R_2 < 0
( (
diverging meniscus R_1 > 0 R_2 > 0
))
converging meniscus R_1 < 0 R_2 < 0
) (
biconcave R_1 < 0 R_2 > 0
| (
plano-concave R_1 = oo R_2 > 0
|)
plano-convex R_1 = oo R_2 < 0
) |
plano-concave R_1 < 0 R_2 = oo
( |
plano-convex R_1 = oo R_2 = oo