Lens-Maker's Formula

The Lens-Maker's formula expresses a thin lens's focal length directly in terms of quantities the person actually grinding the lens controls -- the refractive index $n$ of the lens material relative to its surroundings, and the radii of curvature $R_1$ and $R_2$ of its two surfaces -- rather than in terms of object and image distances. It is obtained by applying the single-spherical-surface refraction formula (Section 14.5) once at each of the lens's two surfaces in turn for an object at infinity (so the emergent rays converge exactly at the focus $F$, i.e. $v=f$), adding the two equations, and eliminating the intermediate image distance exactly as in the thin lens formula's own derivation; the result, for a thin lens of refractive index $n$ surrounded by air, is $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right),$ where $R_1$ is the (signed) radius of curvature of the surface the light meets first and $R_2$ that of the surface it meets second, both signed by the same convention used for $R$ in Section 14.5. This formula is exactly what a lens designer or manufacturer uses in reverse: given the refractive index of the available glass or plastic and a required focal length, it fixes what radii of curvature must be ground onto the two faces of the blank. It also makes clear, at a glance, why a double-convex lens (with $R_1>0$ and $R_2<0$, so $\frac1{R_1}-\frac1{R_2}$ is positive) is always converging when made from a material denser than its surroundings, while a double-concave lens (with $R_1<0$ and $R_2>0$) is always diverging under the same condition, and why the very same pie …