f-number

The f-number describes how steeply a lens focuses light: a lens of focal length $f$ with an entrance pupil of diameter $D$ has $N = f/D$. A small f-number means a wide cone and a fast lens, a large one a narrow cone and a slow lens. It is the working vocabulary of camera lenses and imaging systems; microscope objectives and fiber optics use numerical aperture for the same property.

For a lens focusing a distant object, the two are related by

where $\theta$ is the half angle of the focused cone. The relation is exact for a well corrected lens that obeys the Abbe sine condition, and approximate otherwise. Thus f/2 corresponds to an NA of 0.25, f/4 to 0.125 and f/8 to 0.0625.

What it determines

The f-number sets the size of the diffraction-limited spot independently of focal length. The radius to the first dark ring of the Airy disk is $1.22\lambda N$, 5.4 µm at 550 nm and f/8, and the depth of focus grows as $N^2$, about ±70 µm at the same wavelength and f-number by the common $\pm 2\lambda N^2$ criterion. The image irradiance of an extended scene scales as $1/N^2$, which is why camera f-numbers follow the sequence 1.4, 2, 2.8, 4, 5.6, 8, each step a factor of $\sqrt{2}$ that halves the light.

Working f-number

The nominal f-number applies to an object at infinity. When the lens images a nearby object, the image forms farther from the lens and the cone narrows; the working f-number is approximately $N_w = (1 + |m|)N$, where $m$ is the magnification, assuming a pupil magnification of one. At unit magnification, as in 1:1 relay imaging, an f/4 lens works at f/8, and its diffraction spot and depth of focus are those of f/8.

Measurement

The f-number follows from two measurements, the focal length and the entrance pupil diameter, the latter being the image of the aperture stop seen from the object side, not the physical diameter of the front element. On a bench, the cone angle can also be measured directly by scanning a detector or camera through the focused beam and fitting the growth of the spot with distance from focus.

References: W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008), Ch. 6; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 5.