focal length

Take a magnifying glass out into the sunlight and move it up and down over the ground until the Sun shrinks to the smallest, brightest, hottest dot. The distance from the lens to that burning dot is its focal length. It is the single most important number describing any lens or curved mirror, a compact measure of how strongly it bends light. It answers the question: how far behind this lens (or mirror) does far-away light get squeezed to a point?

The focal length f is the distance from a lens or mirror to its focal point, the spot where rays arriving parallel to the axis (as if from infinitely far away) are brought together. A short focal length means a powerfully bending, strongly curved lens; a long focal length means a gently bending, flatter one. It enters the thin-lens and mirror equations directly through 1/f = 1/d_o + 1/d_i, and for a spherical mirror it is simply half the radius of curvature, f = R/2. Opticians often quote the reciprocal instead: the optical power P = 1/f, measured in dioptres when f is in metres, so a +2 dioptre lens has f = 0.5 m. Converging lenses and mirrors have positive f; diverging ones have negative f.

Focal length shapes everything a lens does: a camera's zoom is really a change of focal length, a long focal length gives a narrow, magnified telephoto view while a short one gives a wide-angle view. Your eyeglass prescription in dioptres is just 1/f of the corrective lens. An honest caveat: for a single thin lens f depends on the colour of the light, because the index of refraction does (dispersion), so blue light focuses slightly closer than red, the origin of chromatic aberration. Real camera lenses combine elements precisely to hold the focal length steady across colours.