There are basically four factors that determine the focal length of a lens: the index of refraction of the lens (n_lens), the index of refraction of the medium which forms the environment of the lens (n_env), the radius of the front surface of the lens (R₁), and the radius of the back surface of the lens (R₂). (The “front” surface of the lens is the surface first encountered by the incoming light.) The equation that relates these quantities to the focal length of the lens is called the lensmaker’s equation,  since it gives the focal length in terms of quantities involved in the manufacture of the lens.

Recalling our discussion of the importance of the signs of quantities associated with optical systems (lenses, objects and images), it should not come as too much of a surprise to find out that there is a quantity that can be either positive or negative that is associated with the (always positive) radius of the surface of a thin lens.  This quantity is called the radius of curvature of the lens surface.  Following our convention, the radius of curvature, which can be either positive or negative, will be denoted by a small letter, r, and the (always positive) radius will be denoted R. 

The radius of curvature is taken to be positive if light, after passing through the surface of the lens in question (either the front or back surface), can pass through the center of curvature of that surface.  Otherwise, the radius of curvature is negative.  (This is very similar to the determination of the sign of the image position, d_i, in which the position is taken to be positive if light can pass through the image point after it passes through the lens, and negative otherwise. The next example will work through determining whether curved surfaces have a positive or negative radius of curvature.)

The lensmaker’s equation is given as follows:

Opticians prescribe corrective lenses for vision disorders in units of diopters (dp), where one diopter is equal to one inverse meter: 1 dp = 1/m.  The dioptric power, d, of a lens in units of diopters, is defined to be d = 1/f, where f is the focal length of the lens in units of meters.  (Be careful—we’ve lots of d’s floating around!)