Refraction through Thin Lenses: The Thin Lens Formula

A lens is a piece of transparent material bounded by two surfaces, at least one of which is curved (spherical); a convex (converging) lens is thicker at its middle than at its edges and, for rays travelling from a rarer to a denser lens material, always bends parallel rays inward towards a real focus on the far side, while a concave (diverging) lens is thinner at its middle than at its edges and always bends parallel rays outward, so that they only appear to diverge from a virtual focus on the near side. Every thin lens (one whose thickness is negligible compared with the object and image distances and with its own radii of curvature) has two principal foci, one on each side, both at the same distance $f$ (the focal length) from its optical centre, because light can pass through a thin lens in either direction. Applying the single-spherical-surface refraction formula of the previous section successively at the lens's first surface (forming an intermediate image) and then again at its second surface (treating that intermediate image as the object for the second surface), and adding the two resulting equations, the intermediate image distance cancels out and the algebra reduces, for a thin lens surrounded by air on both sides, to the compact thin lens formula $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}.$ (Note the sign: for a lens this is a minus between $1/v$ and $1/u$, unlike the plus that appears in the mirror formula -- a direct consequence of the different way the sign convention treats a surface light passes THROUGH versus a surface light reflects FROM.) Applied with the sign convention -- $u$ always negative for a real object, $f$ positive for a convex lens and negative for a concave lens -- this one formula correctly predicts every case: a convex lens forms a real, inverted image (whose size and position depend on how far the object is beyond the focus) for any object farther than $f$, and a virtual, e …