lensmaker formula

The formula (at the bottom of this post) shows for thin, spherical lenses the relationship between shape and power.

Be d center thickness of the lens element.
R_1 and R_2 be the radii of the spheres that describe the surfaces.
Keep the sign conventions for radii in mind, however!

n_0 be the index of refraction of the medium outside the lens and
n be the index of refraction of the lens material.

f be the focallength of the lens and
D be it’s Power, D = \frac{1}{f}

For spherical optical systems holds generally in the paraxial region:

D = \frac{1}{f} = \frac{n-n_0}{n_0}\cdot (\frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-n_0)\cdot d}{n \cdot R_1 \cdot R_2})

Is the surrounding medium air, we get (because n_0 \approx 1) the approximation:

D = \frac{1}{f} = (n-1) \cdot (\frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)\cdot d}{n \cdot R_1 \cdot R_2})

Are the lenses also thin (idealizing d = 0), the formula simplifies to
Lensmaker’s formula:
D = \frac{1}{f} = (n-1)\cdot (\frac{1}{R_1} - \frac{1}{R_2})