Note 15 — Chapter I
Reading edition
Numerical aperture is generally used in the sense in which it was introduced in 1873 by Professor Abbe, on the basis of his theoretical investigations. Numerical aperture represents the ratio between the radius of the effective aperture (p) of the system on the side where the image is formed—more accurately the radius of the emerging pencils measured in the upper focal plane of the objective—and the equivalent focal length (f) of the latter, i.e.,
Numerical aperture = p/f.
This ratio is equal to the product of the sine of half the angle of aperture u of the incident pencils and the refractive index n of the medium, situated in front of the objective. With dry lenses n has therefore the value 1; with immersion lenses it is equal to the refractive index of the particular immersion fluid:
Numerical aperture = n Sin u.
The numerical aperture of a lens determines all its essential qualities; the brightness of the image increases with a given magnification and, other things being equal, as the square of the aperture; the resolving and defining powers are directly related to it, the focal depth of differentiation of depths varies inversely as the aperture, and so forth. (Abbe, “The Estimation of Aperture,” “Journal of the Royal Microscopical Society,” 1881, p. 389.)