Thin lens

A thin lens is modeled with negligible thickness relative to its object and image distances. For real conjugate distances $u,v>0$, it obeys $1/u+1/v=1/f$, with focal length $f$.

For fixed object-screen separation $L>4f$, a converging thin lens focuses at two positions separated by $d=\sqrt{L^2-4Lf}$. Thus $f=(L^2-d^2)/(4L)$. The method finds focal length without needing to locate a thin lens's optical center precisely.

For a thin optical lens in air, surface radii $R_a,R_b$ and refractive index $n$ give $1/f=(n-1)(1/R_a-1/R_b)$. The radii carry the usual signed curvature convention.